In this tutorial we are going to take a look at Power Series. The game with power series is we are trying to plug in different values of x and see what happens. When we plug in a value of x, we get a series just of constants. Then the question is which values of x make that thing converge. So, after introducing the definition of power series we are going to talk more in detail about Radius of Convergence Pattern. Then we will see what is the Interval of Convergence. At the end we are going to do several examples.
Definitions: A power series is a series of the form
(The cn s are the coefficients,
expressions that might involve n, but wont involve x.)
Pattern: The power series
always converges for values of x within some radiusR
around the center a. For a − R < x < a +
R (i.e. |x − a| < R), it is Absolutely
Convergent . For x < a − R or x > a
+ R (i.e. |x − a| > R), it Diverges
. At the endpoints x = a − R and x = a
+ R, it might be conditionally convergent, absolutely
convergent, or divergent.
R is called the radius of
convergence. We can have R = 0 or R = ∞. The
interval a − R < x < a + R (or a
− R ≤ x < a + R, or a − R <
x ≤ a + R, or a − R ≤ x ≤
a + R) is called the interval of convergence.
Hints and tips:
For most power series, you can use
the Ratio Test to find the radus of convergence.
On a few examples, you should use
the Root Test. These examples usually have the form (a function of
However, you can never use
the Ratio or Root Test to check the endpoints, since they will
give you L = 1, which is inconclusive.
You must always check each
endpoint individually, using some test other than Ratio or Root.
Common favorites are the Limit Comparison Test with a p-series,
Alternating Series Test, and Test For Divergence.
A factorial in the denominator
often leads to R = ∞.
Memorize the geometric series
for -1 < x < 1.
You can often derive other power
series from the geometric series by the following methods.
Algebraic manipulations, e.g.
multiplying by x. These wont change the radius of
convergence or whether the series converges at the endpoints.
Substitutions, e.g. replacing x
by 2x or x² . This will change the radius
Derivatives and integrals. These
wont change the radius of convergence, but they might change
whether the series converges at the endpoints.
Other common series that are
worth memorizing (although they can be derived from the geometric
for −1 ≤ x < 1 and arctan
for −1 ≤ x ≤ 1.
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.