In this lesson, our instructor Vincent Selhorst-Jones gives an Intro to Series. Youll review the definition of a series and then discuss why you need notation. Sigma notations, also known as summation notation, and examples of this are gone over in detail. Youll also learn how to reindex by expanding and substituting. The lesson ends with a slide on the properties of sums and then six practice examples.
Given some sequence a1, a2, a3, a4, …, 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:
a1 + a2 + a3 + a4 + …
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 nth partial sum:
a1 + a2 + a3 + …+ an
To compactly describe sums, we use sigma notation (sometimes also called summation notation).
ai is the thing being summed. Terms from the sequence given by ai 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.
Sigma notation can be really confusing the first few times you use it. Check out the video to see an example of how it works and some picture diagrams.
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.
ai = a1 + a2 + a3 + a4 + a5 + …
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 ai and bi be sequences, let c be a constant.
∑i=1nc = c·n
∑i(c ·ai) = c ·∑i ai
∑i(ai + bi) = ∑i ai + ∑i bi
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.