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Introduction to Series
- Given some sequence a_{1}, a_{2}, a_{3}, a_{4}, …, a series is the sum of the terms in the sequence (or a portion of them).
- If the sequence is infinite, we call it an infinite series. It adds all of the terms together:
If the sequence is not infinite or we only wish to add up a finite number of its terms, we call it a finite series. Adding the first n terms of the sequence together is called the n^{th} partial sum:a_{1} + a_{2} + a_{3} + a_{4} + … a_{1} + a_{2} + a_{3} + …+ a_{n} - To compactly describe sums, we use sigma notation (sometimes also called summation notation).
∑
a_{i} - a_{i} is the thing being summed. Terms from the sequence given by a_{i} are added together. This might be a sequence, but is more often an algebraic expression.
- i is the index of summation. It increases by 1 for each "step" of summation. The index can be any symbol, but i is common.
- Above, i=1 is the first value used for the index. This is the Lower Limit of Summation.
- Above, n is the last value used for the index this is the Upper Limit of Summation.
- To show an infinite series (one where the terms keep adding forever), we put `∞' on top of the sigma. This shows that the series has no upper limit and instead continues on forever.
∑
a_{i} = a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + … - Sometimes it is useful to reindex a series (or sequence). We might have a series that has the index begin at one value, but we want it to start at another. However, we can't just change the number for the lower limit, because that would affect the whole series. This means we have to think about how to alter every part of the sigma notation so we can get the index we want without changing the value of the series. There are two main ways of doing this:
- Expand: Write the sigma notation out in its expanded form, then look for how we could rewrite the pattern with our chosen starting index in mind.
- Substitution: How does the old index relate to the new index that we want? Set up that relation, then use substitution to find the new general term and upper limit.
- Sums have various properties that we can occasionally use to our advantage. Below, let a_{i} and b_{i} be sequences, let c be a constant.
- ∑_{i=1}^{n}c = c·n
- ∑_{i}(c ·a_{i}) = c ·∑_{i} a_{i}
- ∑_{i}(a_{i} + b_{i}) = ∑_{i} a_{i} + ∑_{i} b_{i}
Introduction to Series
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
- Intro 0:00
- Introduction 0:06
- Definition: Series 1:20
- Why We Need Notation 2:48
- Simga Notation (AKA Summation Notation) 4:44
- Thing Being Summed
- Index of Summation
- Lower Limit of Summation
- Upper Limit of Summation
- Sigma Notation, Example 7:36
- Sigma Notation for Infinite Series 9:08
- How to Reindex 10:58
- How to Reindex, Expanding
- How to Reindex, Substitution
- Properties of Sums 19:42
- Example 1 23:46
- Example 2 25:34
- Example 3 27:12
- Example 4 29:54
- Example 5 32:06
- Example 6 37:16
Math Analysis Online
Transcription: Introduction to Series
Hi--welcome back to Educator.com.0000
Today, we are going to talk about an introduction to series.0002
In the previous lesson, we introduced the idea of a sequence--that is, an ordered list of numbers.0005
With this idea in mind, we can now discuss the concept of a series, summing up the terms of a sequence.0011
Now, at first, this might seem a little bit series.0018
It is just addition, and we have been doing addition since kindergarten; so why do we need to talk about addition in a special way?0020
But consider how long it would take to add up 100 terms by hand.0025
Could you imagine how much time it would take up if you had to add up 100 different things from a sequence?0029
What if it were a thousand terms, or ten thousand terms?0033
The study of series can make these seemingly colossal summation tasks, having to add up huge numbers of numbers, really, really easy.0037
We can just turn this stuff into being some trivially easy task once we figure out how to talk about series in a deep way.0044
Being able to easily add up lots of numbers has many applications.0051
You will see it in science, engineering, economics, computer programming, advanced math, and many other fields.0054
All of these things benefit greatly from the study of series,0061
from being able to talk about adding up a whole bunch of numbers in an easy way,0064
where we can talk about it in compact notation--all of these things are really important to a variety of fields.0068
So, this is a great thing to study; let's start by defining a series.0073
Given some sequence a_{1}, a_{2}, a_{3}, a_{4}...0078
a series is the sum of the terms in the sequence; it is just adding up the terms,0083
or maybe a portion of them--maybe not the entire sequence.0088
But it is just adding up terms from the sequence.0091
If the sequence is infinite, and we are adding up all of the terms from the sequence, we call it an infinite series.0093
It adds all of the terms together; so I am saying that it keeps going forever.0099
It is a_{1} + a_{2} + a_{3} + a_{4}...going on forever...0104
so plus a_{5}, a_{6}, a_{7}, a_{8}...forever and ever and ever.0110
Since our sequence was infinite, we are just saying to keep adding them forever and ever and ever.0114
That is an infinite series.0119
On the other hand, we can talk about a sequence that is not infinite--or if we only wish to add up a finite number of its terms.0121
We aren't going to add them forever; we are just adding up a finite portion of them--just a number where we can count how many are there.0129
We call this a finite series; adding the first n terms of the sequence together is called the n^{th} partial sum.0136
If we add a_{1} + a_{2} + a_{3}, up until a_{1},0144
but notice: it just stop there--we don't go any farther past a_{n}--then that is the n^{th} partial sum,0149
because we are adding in all of the terms, 1, 2, 3, up until we get to n; so it is the n^{th} partial sum,0154
because it is only a part of our infinite sequence.0160
Consider if we had a sequence to find by the general term a_{n} = n + 2^{n} over 3^{n}.0164
That is 1 + 2 over 3, 2 + 2^{2} over 3^{2}, 3 + 2^{3} over 3^{3},0170
4 + 2^{4} over 3^{4}...what if we wanted to talk about its thirtieth partial sum?0176
In talking about its thirtieth partial sum, we will see very quickly why we need a special form of notation for series.0182
Why is writing this stuff out by hand going to be a real pain?0189
If we wanted to write this out, we would have to first show the pattern that occurs here.0192
We have this pattern that created each one of these various terms.0197
We would have to have that show up in our summation; otherwise, we wouldn't be able to realize what pattern is going on.0202
So, we are going to at least need the first three terms--at least.0208
Then, we could use an ellipsis, that ... , to say that the pattern continues on in this manner.0213
We can not have to write a massive amount, because we can use the ellipsis to say that the pattern keeps going.0218
But we are still going to have to show its stop at the thirtieth term.0223
At a very minimum, we are going to have to write out the first three terms, the last term, and an ellipsis in the middle.0226
We would have to write out (1 + 2)/3 + (2 + 2^{2})/3^{2} + (3 + 2^{3})/3^{3} + ...0232
+ (30 + 2^{30})/3^{30}; that is the thirtieth partial sum, like we were talking about.0240
But that is a lot of stuff to write out; that is a lot of writing for a fairly simple series.0246
This isn't even that complicated; but if we had to write this on multiple lines, as we worked through a problem--0250
if we had to keep talking about this over and over as we did steps in a problem--your wrist is going to hurt after doing one of these problems.0256
And you are going to have to do a bunch of problems about summation.0261
This is why we need some sort of notation: we need notation to make it easier to compactly describe a series.0263
We don't have to write out this really, really long thing every time we want to talk about some summation from a sequence.0271
We need some easy way to be able to do it in a short way.0278
Enter sigma notation: to compactly describe sums, we use something called sigma notation.0281
It is also sometimes called summation notation.0287
Why is it called sigma notation?--because it uses the uppercase Greek letter sigma.0290
Sigma is this right here; if you are drawing it by hand (this is a computer-drawn picture of sigma,0295
made by a computer typesetting program), I would recommend writing it like this,0302
where you have these little vertical hooks on either end; and then it is just kind of a capital M on its side.0312
Just write it out like that; and that is great.0319
If you are feeling lazy, you can end up just sort of writing it like that, as well.0323
But it helps to write it like this, so that we can clearly see what it is.0326
But if you are doing a lot of problems that way, don't worry if it ends up getting not quite absolutely perfect each time,0329
because you will be able to recognize it from other things that are about to be seen.0335
All right, let's see how we use sigma notation.0338
Let's look at the anatomy of a series in sigma notation.0342
The first thing to look at is the thing being summed: the terms from the sequence given by a_{i}, this thing to the right of the sigma, are added together.0346
This is what will be added together over and over with each step as we add up:0356
the first thing, and then the second thing, and then the third thing, and so on.0360
It might be a sequence, but it is much more often going to be an algebraic expression--something like 3 times i plus 10 or 2 to the i or i!,0362
something that we could plug in a number and be able to churn out a value.0373
You will often see algebraic expressions; but once in a while, you will see a sequence like a_{i}.0376
Next, the index of summation; i is the index--it increases by 1 for each step of summation.0382
We do the first thing, and then we add the second thing in the next step;0391
and then we add the next thing in the next step; and we add the next thing in the next step, and so on, and so forth.0394
So, every time we step forward, the index will click up by 1.0399
We start at some value, and then we go to the value one above that, then the value one above that, then the value one above that, and so on.0403
So, every step, it will increase by 1.0409
The index can be any symbol; but i is very common, and that is what we will end up using for the most part in this course.0412
So, that is the thing that ends up changing; and notice how i will generally occur over here in the series, as well.0419
That is saying that this is the thing that changes; this is what will change with each step--this little thing right here.0425
The lower limit of summation: this is the first value used for the index.0432
This is where the series starts; in this case, since i = 1, our first value would be plugging in a 1 for the i.0436
The upper limit of summation: this is the last value used for the index, where the series ends.0444
We would step up until we eventually got to some value n.0450
All right, let's see it in action: we have this pictorial summary of how it works.0454
This is a great picture; so check against this, if you get used confused in a later point; this is a great slide right here.0458
Let's see it in action: we have sigma...it's i = 3 for the starting index, and then 7 is the upper limit, and we have 2i - 1 as the actual expression.0464
The very first thing: we have the i right here, and we start at i = 3.0475
So, the first value that we plug in for our i is 2 times 3, because our lower limit of summation was that 3.0479
So, 2 times 3 minus 1 is the first thing.0486
Then, we go on to our next step: plus...we increase our index by 1.0489
Our index started at 3, so it goes from 3...plus 1...to 4; so 2 times 4, minus 1.0493
Then, we go on to our next step; it will be 2 times...we were at 4 in the last step, so it increases to 5; so we are at 4 + 1 is 5.0500
2 times 5 minus 1, plus...our index increases by 1 again, so 2 times 6 minus 1, plus...0508
and now, finally, we have just hit our final, upper limit.0516
So, at this point, we stop; this will be the last one, once we click up to this final upper limit for how high our index goes to.0520
That is the last one in the series; so it is 2 times 7 minus 1.0528
And at this point, we could simplify it out if we wanted to.0533
We could get a value from this; but that is also a good way to see how it expands.0535
And that is all we are looking for right now--seeing how it expands.0539
But if you wanted to, you could simplify each one of these.0541
And you could also combine them and be able to get a value for that series.0543
So far, we have only seen how to use sigma notation for finite sums--that is to say, partial sums, not sums that are going on forever.0548
Since they have all had some upper limit of summation, something on top of this sigma, they have all had to eventually stop.0555
They have had some top limit to how far they step out.0561
If we want to show an infinite series, one where the series' terms keep adding forever and ever,0564
we simply put an infinity symbol on top of the sigma.0570
This shows that the series has no upper limit, and instead continues on forever.0573
So, if we had a sigma with i = 1 on the bottom, our index is i, and our lower limit is 1,0577
and our upper limit is infinity, that says really "just keep going."0583
It is not that we actually get to infinity and stop; we can't ever get to infinity.0586
Infinity is just the idea of going on forever; so we start at 1, and then on to 2, and then on to 3, and then on to 4, and then on to 5, and then on to 6...0589
And it just keeps going forever and ever.0598
So we would have...since it is a_{i}, we would have a_{1} first, because it was a 1 for our lower limit,0600
then a_{2}, and then + a_{3}, and then + a_{4}, and then + a_{5}...0605
and it will just keep going on forever, because it is an infinite series that we are working with.0609
If we wanted to see in specific, here is an example that uses specific numbers.0614
We have some series with i = 1 on the bottom; it is an infinite series, because of the infinity on top.0618
We have 1/2 to the i; so our first i is 1; that is 1/2 (is our first term, one-half), plus 1/2...0623
our next step with the i will be now at a 2, so 1/2 to the 2, 1/4...and then our next step would be i at 3;0633
so 1/2 to the 3; that is 8; at our next step, i will be at 4, so that is 1/2 to the 4, so 1/16.0641
And then, i will be at 5, so it is 1/2 to the fifth, so 1/32; and that pattern will just keep going.0648
We will keep adding them on forever and ever.0653
Sometimes it is useful to re-index a series (or a sequence, sometimes).0657
We might have a series that has the index begin at one value, but we want it to start at another value.0662
For example, we might want a transformation like this one here, where we have, in our original sequence,0667
i = 7 as our starting index, and it has some upper limit, and it has some expression that we are actually working with.0672
And we want to transform it to k = 1; we want our starting index to be k = 1.0678
And we are going to, of course, also need some upper limit and something here.0684
Now, of course, we can't just change the number of the lower limit.0688
If we just went and changed the lower limit, that would affect the whole series.0691
We have to pay attention to how the thing being summed, and our upper limit, end up making the transformation, as well.0696
They are going to change over the process; they are not going to be the same thing on the left side and the right side.0703
They are going to end up becoming different things; otherwise we would have altered the whole thing by starting...0710
we start at one starting place, and then we just change the starting place, and we don't do anything else;0715
well, if you start in a different starting place, you are going to have different values coming out.0718
So, since we want to end up having the same value come out of our two ways of talking about it,0721
we have to alter every part of the sigma notation, so that we can get the index we want without changing the value that comes out of it.0725
You are probably asking why we care; if we have it written in one way, why not leave it that way?0734
Well, there are a bunch of reasons to do it.0738
For example, if you are working on a proof, it can sometimes help to re-index it.0740
But specifically to this course, a lot of formulas that we end up working with will be given in the form of something with an i = 1 on the bottom.0743
So, since a lot of our formulas will have i = 1 on the bottom, we will have to start at this index of something = 1,0752
some symbol = 1, to be able to work with some of the formulas we have.0758
A lot of the formulas that you end up seeing are in this format.0762
So, it is really helpful to be able to re-index, so we get to the format that we already have a formula for,0764
as opposed to having to figure out an entirely new formula for some new, different index.0768
So, how do we actually re-index?0773
The most essential way is to expand the sigma notation into a written-out series,0775
then see how we could rewrite the pattern with our chosen starting index in mind.0779
So, for example, if we want to go from i = 7 to k = 1, we can just start by...0783
a series is just a shorthand way of writing out something + something + something...+ something.0791
And even that is a shorthand way of writing out every single term.0797
So, we can expand it into that format; if it is i = 7 first, then we would have 3(7) - 16;0800
the next thing would be + 3 times 1 step up to 8, minus 16; plus...it would continue in this form.0807
And 3 times...our final upper limit is 22; so 3(22) - 16; great.0815
If we want, we can figure out how to write this sigma over here, with just that in mind.0822
But we can also simplify things to see if we can see another easier-to-see pattern.0827
So, 3(7) would give us 21, minus 16, plus...3 times 8...16, 24...minus 16 + ... 3(22) is 6, minus 16...0831
so, 21 - 16 is 5, plus 24 - 16 (that is 8), plus...plus 66 - 16 (that is 50).0845
We look at this, and we might realize, "Oh, what is doing is going up by 3 each time."0854
And that makes sense, since we started at 3i.0858
It is going up at 3 each time; if that is the case, and we want to get this k = 1, then that means0860
that we know that here, we are at k = 1; here we are at k = 2; here we are at k =...0865
we will actually leave that as a question mark right now, because we don't quite know yet.0870
We will talk about that in just a moment--what number we are at; we will see why.0873
k = 1; if it is + 3 each time, we are going to want some 3k + some number.0877
So, if we are at 5 here, then that would be 3k + 2, because 1 for k...3 times 1 plus 2 does get us 5, so that checks out.0882
Here, with 3k + 2, it works, as well; so 3k + 2 is here; we plug in 2; 3(2) is 6, plus 2 is 8; that checks out, as well.0889
So, it looks like it is going to be some 3k + 2 that will be here; we will have 3k + 2 for that part right there.0897
Now, what is the upper limit of summation going to end up being?0905
Well, if we are at 3k + 2, and it equals 50, we have to figure out what value would end up coming out here.0908
So, k = ?; we could solve for this: 3k = 48; k...divided by 3 now on both sides...we get 16.0915
So, we know that our upper limit is going to have to be 16.0922
Another way of doing it is to notice what the difference is between our upper limit and our lower limit.0925
In our original, we had 22 as our upper and 7 as our lower; so, 22 - 7 means 15.0930
Oops, I'm sorry; not 15...oh, it is 15; 22 - 7 comes out to be 15.0936
Over here, we know that we are going to end up having that difference of 15, as well.0944
Since we start at k = 1, 1 plus that same 15 (because we are going to have to have the same number of steps,0949
however we phrase it) comes out to be 16, which is the same thing we figured out that it has to be to work there.0958
With that in mind, we now know that our lower limit with our index is going to be k = 1,0965
because that is what we wanted to figure out in the first place.0972
Our top, our upper limit of summation, will be 16; we have figured out two different ways that that has to be the case,0974
because 16 - 1 is 15, which is the same as 22 - 17; or alternately, we can solve for it0980
using the formation that we came up with for that general term.0985
And then, 3k + 2 is what is actually going in that is being added on each term.0988
And so, that is one way of doing it.0994
We worked this out by expanding first; expanding is a very specific way to see it.0995
It is a great thing to do if you get confused by the problem that you are working on.1000
There is another way to do this, though; we can also re-index by thinking in terms of substitution.1004
How do our old index and the new index we are creating relate to each other?1008
We can think in terms of substitution: consider, once again, if we wanted to convert from i = 7, with all of the same thing, to k = 1.1012
Well, since we have i = 7 and k = 1 both at their starting places, how is i related to k?1020
Well, i is the same thing as k + 6; 1 + 6 gives us 7.1026
So, we now have this relationship where i is equal to k + 6.1031
With this in mind, we can substitute to figure out what the upper limit is, and then to figure out what the general term is.1036
To figure out the upper limit, we know that i = 22 is what we are going to plug in, since that is its upper limit.1043
And then, we are going to...over here, we know i = k + 6, so we plug in 22 for i: we have 22 = k + 6.1051
Subtract 6 on both sides, and we get 16 = k for its upper limit.1062
So, at this point, we can now write that our sigma is going to end up being...upper limit of 16;1072
the lower starting index is at k = 1; now, what is it going to end up being?1080
Well, here it was 3i - 16; so we have 3; what is our i in terms of k?1085
Well, i is k + 6; and then, we continue on with the rest of it...minus 16.1093
We have substituted out the i here for the k + 6 here, into what i was in our initial version for the sigma notation.1100
So, at this point, we can expand this and work this out, and simplify it if we want.1111
We could also just leave it as it is, and it would be just fine.1114
The upper limit and lower limit and our index--they will all stay the same throughout.1117
We work this out: 3 times k plus 6 gets 3k +...3 times 6 is 18; minus 16; and so, we get 16 for our upper limit; k = 1 for our index and lower limit.1121
3k + 18 simplifies to 3k + 2, which is exactly what we had the first time when we did this through expanding it.1135
So, substitution and expanding both work the same way.1146
Substitution is probably a little bit faster and easier, but it is a little bit more complicated to see what is going on, to really understand.1148
The important thing is to figure out how your i...whatever your initial symbol and the new symbol that you are changing to are...1153
how they are related to each other; and you probably start with the lower limit,1159
and then ask what your upper limit will have to be, to have that same relationship between old index and new index.1162
And then, what is your expression going to end up being, if you just plug in your connection between the two indexes?1168
All right, working with series, there are various properties that we can occasionally use to our advantage.1177
Let's look at some properties of sums: below, let a_{i} and b_{i} be sequences.1181
They are things that are liable to change; but c will simply always be a constant.1187
Our first one is that the sum from i = 1 to n of c is equal to c times n.1191
Remember: c is just a constant.1197
Why is this the case? Well, remember: if we had the sum of i = 1 to n of c,1198
well, c doesn't change any time, because it is a constant; it isn't affected by the index.1206
So, it is just going to be c, plus c, plus c...plus c, that many times.1210
If we have that, how many times did it show up?1216
Well, we went from i = 1 up until n, so 1, 2, 3, 4...we count up to n; we have a total of n terms in there.1219
If c adds to itself n times, then that is just c times n.1227
So, that is equal to c times n; and that is where we get that property for how sums work.1233
If it is just a constant being added through a series, we can just multiply it by the number of terms in that series.1239
Next, we have the summation of some summation...and this will end up working for any summation limit;1246
the only thing that we have to care about is the index.1253
You can start with any lower limit and any upper limit, including an infinite sum; it is still going to end up working out the same.1255
So, c times a_{i}...we can do this where we pull out the constant, and we bring it out to the front.1261
And it is c times the summation of a_{i}.1267
So, why is this the case? Well, notice: our first term would be c times a_{1}.1270
Plus...our next term would be c times a_{2}; plus...our next term would be c times a_{3}.1276
And it is going to continue on in this pattern.1283
It might end; it might not end; we will just leave it as dots there.1285
But notice: we have a c on each one of our terms.1288
So, if we want, since we have a c here, a c here, a c here, and that thing is going to end up continuing1291
(there is going to be a c on each one of these, because we see it is c times a_{i}),1295
we can pull out all of the c's; we pull them out, so it is c times a_{1} + a_{2} + a_{3} + ....1299
So, it is c multiplied against that entire series; it is not going to have any difference in how we do it.1310
So, this entire series, here to here--we can think of it as c times a new series of just the a_{i}'s changing.1315
And that is how we end up getting the same thing here.1323
The final idea: if we have addition, a_{i} + b_{i}, well, we can end up breaking that into two separate series, added together.1327
And once again, this would end up working with any limit; so that is why we don't have a lower limit and don't have an upper limit--just an index.1335
It is because this will work with any limits whatsoever, even an infinite series.1340
We could write this...if it is series of i, a_{i} + b_{i}, well, that is going to be a_{1} + b_{1},1346
plus a_{2} + b_{2}, plus a_{3} + b_{3}...1353
And it is just going to continue on in that format.1360
Well, order of addition doesn't matter; a_{1} + b_{1} + a_{2} + b_{2}1363
is the same thing as a_{1} + a_{2}, then plus b_{1} + b_{2}.1368
So, what we do is order all of our a_{1}'s first; we will have all of the a_{1}'s in our series show up first.1371
And then, we will add on our b_{1}'s: b_{1} + b_{2} + b_{3}...1381
And that will also go on the same as it would have on its own.1388
So, at that point, what we have done is just re-ordered how we are looking at the series, since we have expanded it,1393
and now we have re-ordered the way we are looking at it.1397
But we haven't changed anything; so at this point, we can just pull it into two separate series, i...1399
so, we will have a_{i} here, plus...and then the b_{i}.1404
There is the a portion of the series and the b portion of the series.1410
So, we can either have them intermixed together, or we can spread them apart and work on each of them on its own.1413
And that is how we end up having this final property.1419
All right, we are ready for some examples.1422
The first example here: Given the sequence below, find the third partial sum.1424
If we are going to find the third partial sum, that is as easy as just adding up the first three terms.1429
That is 0, 3, and 8; 0 + 3 + 8; we end up getting 11; it's done; it was easy.1433
The next one: If we are looking for the seventh partial sum...well, so far, we only have 0, 3, 8, 15, 24...1443
so we are going to have to figure out what the pattern will continue on to.1448
Our first step is to figure out how this pattern ends up working.1452
We look at this through addition: 0 to 3, 3 to 8, 8 to 15...well, the addition is changing each time.1455
So, we could think in terms of some recursive relationship, but that is not really going to make it easier.1460
Multiplication? Multiplication doesn't really work, either.1464
But we look at this for a while, and we might realize that this looks kind of like 1, 4, 9, 16, 25...1466
0, 3, 8, 15, 24...that is just - 1 on each one of the terms.1476
So, we can think of this as being, in general, n^{2} - 1.1480
If that is the case, then we can figure out what the next terms are.1486
If it is n^{2} - 1, the next term to follow is going to end up being 36 - 1, because we are at 5 here; 5^{2} - 1 is 24.1489
So, the next will be 6^{2} - 1; that is going to get us 35.1497
The next one will be 7^{2} - 1, 49 - 1, or 48.1502
At this point, the seventh partial sum will be adding up these first seven terms.1507
So, 0 + 3 + 8 + 15 + 24 + 35 + 48...we toss that all into a calculator, and we end up getting 133; there we go.1511
The next one: Expand the below sigma notation into a series of terms; add them together using an ellipsis.1530
We break it apart into the thing where we are not using sigma notation.1536
We are just writing it as one term after another, with some pattern occurring.1539
So, what would our very first term be?1542
Our very first term would start at i = 3; so we plug in a 3 for our i first; that would be 3! over 7^{3}.1544
Plus...our next term would end up being...we click up one, from 3 to 4; that will be 4! over 7^{4}.1554
Plus...our next one: we click up another one to 5! over 7^{5}, plus..., plus...we are going to end up finishing here at 15.1564
We could also start with something before 15; for example, we could write out 14!,1574
because that would be in the expansion: 7^{14}, plus 15! over 7^{15}.1579
There we go; we have managed to write the whole thing out, using ellipsis.1589
We have expanded the sigma notation into a series of terms.1591
And we don't have to necessarily write out all of these.1596
We could probably get away with not writing it.1599
We could certainly get away with not writing this one right here, the 14!/7^{14}.1601
And we might even be able to get away with not writing the 5!/7^{5}.1605
It helps us see the pattern, but it is not absolutely necessary.1609
But by including more terms, sometimes the reason we want to expand things is so we can manipulate how the terms work.1612
So, sometimes it is useful to include more things at the start and the end, so that we can end up seeing how things are interacting.1618
It will sometimes allow things to work out better.1625
And we will see that sometimes, when we are working in proofs.1627
Example 3: Condense the sum below into a series expressed using sigma notation.1631
Notice: it is an infinite series.1635
The first thing: let's notice how these connect to each other; how do we get 1/2 - 1/4 + 1/8 - 1/16?1637
Well, we can break this into a sequence, first, that just has a series going on.1643
So, let's look at this underlying sequence; how do we get from 1/2 to -1/4? Well, we multiply by -1/2.1647
How do we get from -1/4 to positive 1/8? We multiply by -1/2.1653
How do we get the next one? We multiply by -1/2.1658
So, with this in mind, how can we create a general term a_{n}?1661
That is going to be a_{n} =...since we are multiplying by -1/2 each time, it will be -1/2 to the...1665
is it going to be to the n? Well, here we are going to end up starting with something else, actually.1672
It is n - 1, because here is n = 1; we want to not have anything at the beginning, which we could write as times 1/2... so -1/2 to the n - 1, times 1/2.1677
Alternatively, another equivalent way to write this out: we could write this as -1 to the n - 1 over 2 to the n - 1, times 1/2.1687
The 2's in the denominator will end up compacting together.1699
And we could also write this as -1 to the n - 1, over 2 to the n.1701
Both of these are just fine ways to write it.1710
We will end up getting the same thing, whether we write it using this general term, or we end up writing it with this general term.1711
We will end up getting the same series.1718
What is our first term? We decided to set it at some n = 1; let's use an index of i,1720
just because it is what we are used to, although we could use n.1724
Sometimes, you might have a little bit of confusion, since we are used to talking about n^{th} terms with n.1727
So, it might be weird to use an index of n; but you will often see it used, as well.1731
I like i, so I am going to use i.1734
i = 1, because that is our first location; what do we go up to?1737
Well, it is an infinite series, because it goes on forever, and we are never told that it stops.1740
So, we use an infinity symbol on top; and then, let's do this one first.1745
We could write this as -1/2 to the n - 1, times 1/2; and that whole thing has the series applied to it.1749
Or equivalently, this would also be equal to the series--the same upper limit of going on forever, and the same starting index.1759
We could also write this as -1 to the n - 1 over 2 to the n.1770
Either way we end up working it out, both of these are just fine.1775
They are going to give us the exact same answer; they are just two different ways to write it out.1780
Depending on the specific problem, it might be useful to write it one way or the other.1783
But any teacher, if this was just the question, should accept both of these.1787
All right, calculate the value of sigma with a lower limit of 0 and an upper limit of 4.1791
These are indexed where the expression is v! - 5v.1796
The first thing we do is plug in v = 0 for our first 0.1800
Our very first term is going to be 0! - 5(0).1805
Our index is v, so that is the thing being swapped out.1810
Plus...we step up our index to the next level, 0 up to 1 now; so 1! - 5(1).1813
Plus...the next thing, stepping up from 1, is 2; so 2! - 5(2).1822
Plus...the next thing, from 2 to 3: 3! - 5(3); and then, plus...3 to 4...4! - 5(4).1829
Finally, at this point, we notice that that is our upper limit, so we stop.1841
We step from our lower limit, 0, 1, 2, 3, 4...our upper limit, so that is where we stop.1845
And we do each of the steps in between and add them all together.1851
So now, it is just a matter of simplifying to actually get the value.1853
What is 0!? Remember: 0! is simply defined to be equal to 1.1856
All of the other numbers, factorial, are that number, times each of the positive integer numbers underneath it.1861
But 0! is just simply defined to be 1.1868
So, 0! gets us 1; minus...5 times 0 is 0, plus 1! is just 1 times itself; so 1 minus...5 times 1 is 5, plus 2!...1872
2 times 1 is 2, minus 5(2) is 10, plus 3! is 3 times 2 times 1, so it is 6; minus 5(3) is 15;1883
plus 4! is 4 times 3 times 2 times 1, 24; minus 5(4) is 20.1893
We work this out; we have 1 here, plus -4, plus -8, plus -9, plus +4.1901
We see that we have a +4 here and a -4 here; they cancel out.1912
1 + -9 gets us -8, so we have -8 and -8, or -16, once we simplify the whole thing out.1917
Great; the fifth example: Re-index the series below so that it starts at the index k = 1.1924
We are looking to swap the index i = 5 out for k = 1, but have the exact same value come out of the series.1930
Our sigma notation will change; the upper limit will end up changing, and the expression here will end up changing to some extent,1937
so that we can achieve this without affecting the value of the series.1943
We talked about two different ways to do this when we went through the lesson.1947
The first way we will approach this is through expanding it.1951
We will start by doing it through expanding it; and then later, we will look at doing it through substitution.1957
There are two alternative ways; whichever one makes more sense to you, that is the one that you probably want to end up using for your own work.1964
All right, we want to start by expanding this.1969
If we are going to expand i = 5, 3^{7 - i}, that is going to be...our first thing would be 3^{7 - 5}, so 3^{2}.1972
Plus...then the next would be 6 for our i; so 3^{7 - 6} would be 3^{1},1981
plus...then at 7, we will have 3^{0}, plus...plus...and then finally,1990
when we get up to an upper limit of 20, 7 - 20 is -13, so it will be 3^{-13}.1997
OK, that is what we see that we end up getting out of this: 3^{2} + 3^{1} + 3^{0}...+ 3^{-13}.2004
So, we see that the top steps down each time.2011
We want it to figure out how we can end up showing this.2015
We start with k = 1 for this first slot; use k = 2, k = 3...and here is k = we-don't-know-what-yet.2017
We can actually figure out that it is going to have to be...20 - 5 is 15, so 1, our starting lower limit here, plus 15 would be 16.2026
We know it will end up having it come out to 16; but let's work through it the other way.2034
So, what we have here...it looks like we could write this as 3^{3 - k}.2038
3^{3 - 1} would get us 3^{2}; the next one, 3^{3 - 2}, would get us 3^{1}; that works out.2045
At k = 3 on our third term, we would have 3^{3 - 3}; it would be 3^{0}; that ends up working out.2054
So, if we are going to have 3^{-13} = 3^{3 - k}, we are now going to know that -13 has to be equal to 3 - k.2059
So, we have k = 16 as our upper limit, as well; great.2069
That ends up making sense; we can also figure it out, once again, from:2076
our starting upper limit was 20; our starting lower limit was 5.2079
So, that means that there is a difference of 15; so if we have k = 1, then 1 + 15 has to come out to be 16, as well.2083
So, that is going to be our upper limit; there are two ways of looking at it.2093
Either way, just make sure that you are careful with this sort of thing.2095
It is easy to get confused the first few times you work with it.2097
So now, we know what our general term is; it is 3^{3 - k}.2100
We know what our upper limit is, and we know what our index and starting term is, and lower limit.2103
So, we can write the whole thing out: we have 16 as our upper limit; k = lower limit of 1; 3 to the 3 - k; great.2109
So, that is our answer; and that is doing it through expanding as our method.2120
Alternatively, we could do this through substitution.2125
We could have also set this up by noticing that we have i = 5 here, and we want to start at k = 1.2128
So, at the first place, we have i = 5; and then, at our second place, we have k = 1.2133
So, if we have i = 5 here and k = 1 here, then we can see that i is equal to k + 4.2139
So, if i is equal to k + 4, in general (this is just going to hold true in general), then what about our upper limit?2146
Our upper limit: it is going to have to end up being the case that when i is equal to 20...what will our k end up being?2154
We will have 20 = k + 4, which means 16 is equal to k for the upper limit.2160
16 = k for our upper limit; now we just plug in through substitution.2167
So, 16 is our upper limit; we just figured that out; k = 1 is our index and lower limit.2172
So, it is going to be the same thing we started with, 3^{7 - i}, so 3 to the 7 minus...2178
but we are not going to use i; we are going to use i = k + 4.2186
So, we are going to swap out this i here for k + 4.2190
We substitute that in, and we have k + 4.2194
We can now work this out and simplify it, since it is the upper limit and lower limit and index...2199
none of those will change; we are just simplifying what is on the inside.2205
3 to the 7 - (k + 4), so minus k and minus 4...7 - 4 gets us 3, and minus k gets us 3 - k in the whole.2208
So, we end up getting the exact same thing, either way we end up approaching this: through substitution or through expanding.2218
We can end up re-indexing the series, so we can have it change at a different starting index.2224
They both work fine; whichever one makes more sense to you, that is the one I would recommend using.2229
All right, the final example: Calculate the value of the series below; use summation properties to make the math less tedious.2233
We could just do this by hand; we could say, "All right, we have 3 times 1 minus 5, plus...plus...all the way up to 3 times 15 minus 5."2240
OK, and then we could calculate what 3 times 1 minus 5 is, and what 3 times 2 minus 5 is, and what 3 times 3 minus 5 is.2252
And we could do this by hand or through the calculator; but that is kind of a pain.2258
That is going to be a lot of writing; that is going to be a lot of calculation.2262
Luckily, there are some clever ways to use the summation properties that we learned early to make this at least...2265
not easy...not fast (it will be easy), but less tedious, so we have to do less writing.2270
How can we do this? Well, first, remember: we can split based on addition.2276
We can see this as 3i + -5; and we can split this into i = 1; the limits to our summation will never end up changing,2280
but we can split this into a 3i +...the same sum over here...15i = 1 on -5; cool.2290
We also were allowed to pull out constants; we have this constant of a 3 here; we have this constant of a -1 here, effectively.2299
We can pull those out front; we can now write this as 3 times the summation of 15, i = 1 of i, plus...2306
Now, I am sorry...plus -1, so let's just write that as minus...summation...15...i = positive 1 (I got confused by the negative) on 5; great.2319
So, 3 times...well, there are no cool properties that we have learned yet, although we will learn in the very next lesson,2330
how to easily sum this one up; but we would have 1 plus 2 plus 3 plus...plus 15.2338
And over here, we will have minus...well, we could do 5 + 5 + 5...but it is just going to show up 15 times.2345
We have i = 1 to 15; we talked about this before.2351
Since a constant is just showing up 15 times, that is going to be 15 times the constant.2354
So, it will be 15 (the number of terms that are there) times the constant that is showing up over and over, so 15 times 5.2359
We can now work out with a calculator 1 + 2 + 3 +...up until 15.2366
That ends up working out to 120; so we have 3(120) - 15(5) is 75; 3 times 120 is 360, minus 75; that comes out to equal 285, and there is our answer.2372
We still had to do a little bit of writing this out.2389
The slowest part here is going to be adding by hand: 1 + 2 + 3 +...to 15, or by using a calculator.2391
But either way, that is much better than having to multiply numbers, and then subtract 5, and then add that, each time, over and over.2399
So, we can use these summation properties to split things up into ways that make them easier to work with.2406
All right, cool; in the next lesson, we will end up looking at our first specific kind of sequence and series, arithmetic.2411
And then later on, we will work with geometric, which will let us apply these ideas about series into one specific thing,2416
where we can actually start creating some formulas to make things really easy and fast.2421
All right, we will see you at Educator.com later--goodbye!2424
1 answer
Last reply by: Professor Selhorst-Jones
Fri May 9, 2014 6:11 PM
Post by abendra naidoo on May 8, 2014
hello,
I came across this problem in my reading and I am stumped as to how the answer can have 2 terms with a plus in between:Find the exact sum of the first 14terms in the geom. sequence sqrt.2,2,2sqrt2,4Ã¢ï¿½Â¦
The answer given is 254+127sqrt2.
Using formula Ssubn=Asub1(1-r^n)/1-r --- I can't see how you end up with a plus sign in the answer?
Thanks.