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You CAN Ace Calculus

17calculus > infinite series > power series

Power Series

Difference Between Power Series and Taylor Series

Taylor series are a specific case of Power series where the constants ( usually functions of n ) are related to the derivative of the function. Many of the same techniques that work for one will work for the other.

A Power Series is based on the Geometric Series using the equation \(\displaystyle{ \sum_{n=0}^{\infty}{ar^n} = \frac{a}{1-r} }\) which converges for \( |r| < 1 \), where r is a function of x. We can also use the ratio test and other tests to determine the radius and interval of convergence.
Power Series are discussed on this page.

A Taylor Series is based on the recurring formula
\(\displaystyle{ \sum_{n=0}^{\infty}{\frac{f^{(n)}(a)}{n!}(x-a)^n} = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + . . . }\)
We use the Ratio Test to determine the radius of convergence.
Taylor Series are discussed on a separate page.

The idea with power series is to use the fact that with a geometric series, we know it converges under certain circumstances AND we know what it converges to. So what we do is use algebra and, sometimes calculus, to write a function in the form of a geometric power series.

The geometric series equation is \(\displaystyle{ \sum_{n=0}^{\infty}{ar^n} = \frac{a}{1-r} }\). We will have equations where r is a function of x.
We then match the values of a and r to get the series form of the function. We also need to specify the values of r for which this series converges, i.e. \( \abs{r} < 1 \).

Why would we do this? Well, one reason is that we can convert a somewhat more complicated function into a polynomial (which happens to be infinite) and it is much easier (almost trivial) to find derivatives and integrals of polynomials.

Sometimes, you can just use algebra, to get the equation in the right form. But other times, you need to do a little more than basic algebra. If you can either take the derivative or integrate the original function to get something in the form \(\displaystyle{ \frac{a}{1-r} }\), then you can do the opposite (integrate or take the derivative) of the sum to get the original function. I'm sure by now you need an example to get your head around this. Let's watch some videos that explain these ideas in more detail with examples.

Here are several very good videos introducing and explaining power series. It will help you to watch all of them, in this order, so that you can get several different perspectives on the same topic.

This first video introduces power series and its relationship to Taylor/Maclaurin series. He also uses the example \(\displaystyle{ \sum_{ n=0 }^{\infty}{ \frac{1}{2^n}x^n } }\) to talk about under what conditions this series converges. If you have time to watch only one video, this is the one to watch.

Dr Chris Tisdell - What is a power series?

This next video clip is more theoretical than the previous one but it will help you understand why power series work.

MIT OCW - Lec 38 | MIT 18.01 Single Variable Calculus, Fall 2007

The next video is very practical, giving you the basics and showing you how to work with power series.

PatrickJMT - Power Series Representation of Functions

Manipulating Power Series

Okay, now that you know how to set up power series, you can do all kinds of manipulations of them to find power series of other functions. In this video, he discusses how to manipulate power series, by taking derivatives and integrating, in order to find power series representation of functions that are not already in geometric series form.

Dr Chris Tisdell - What is a power series?

Finding a Power Series Using The Binomial Series

The binomial series can be used to expand a special class of functions into power series.

If k is any real number and \( |x| < 1 \), then
\(\displaystyle{ (1+x)^k = 1 + kx + \frac{k(k-1)}{2!} x^2 + \frac{k(k-1)(k-2)}{3!}x^3 + . . . = \sum_{n=0}^{\infty}{ \left( \begin{array}{c} k \\ n \end{array} \right) x^n } }\)
where \(\displaystyle{ \left( \begin{array}{c} k \\ n \end{array} \right) = \frac{k!}{n!(k-n)!} }\)

You can also use Taylor Series Expansion to expand these kinds of functions. There are a couple of practice problems below that use this series.

Radius and Interval of Convergence

Radius and Interval of Convergence

This section is part of the discussion found on the radius and interval of convergence page, where you will find more information, videos and practice problems.

Once the Taylor series or power series is calculated, we use the ratio test to determine the radius convergence and other tests to determine the interval of convergence. Without knowing the radius and interval of convergence, the series is not considered a complete function (This is similar to not knowing the domain of a function. As we say in the page on domain and range, a fully defined function always contains information about the domain, either implicitly or explicitly stated.) In this case, we can't leave off information about where the series converges because the series may not hold for all values of \(x\).

Using The Ratio Test

The ratio test looks like this. We have a series \( \sum{a_n} \) with non-zero terms. We calculate the limit \( \displaystyle{\lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right|} = L} \).
There are three possible cases for the value \(L\).
\( L < 1 \): The series converges absolutely.
\( L = 1 \): The ratio test is inconclusive.
\( L > 1 \): The series diverges.

So we use the first case ( \(L<1\), since we want convergence ) and we set up the inequality \( \displaystyle{\lim_{n \to \infty}{\left| \frac{a_{n+1}}{a_n} \right|} < 1} ~~~~~ [ 1 ]\)

Key: Do not drop the absolute values.

Radius of Convergence

To find the radius of convergence, we need to simplify the inequality [1] to the point that we have \( \left| x-a \right| < R \). This gives the radius of convergence as \(R\). This seems very simple but you need to be careful of the notation and wording your textbooks. Some textbooks use a small \(r\). Some textbooks ask for the ratio of convergence in which case you need to give the answer as \( \rho = 1/R \). This has always seemed kind of strange to me but there must be some reason for it. However, on this site, whenever we talk about the radius of convergence in this context, we will use \(R\) as defined above.

There are three possible cases for the radius of convergence.

\(R = 0\)

series converges only at the point \(x = a\)

\( 0 < R < \infty \)

series converges in the interval

\(R = \infty \)

series converges for all \(x\)

The interesting thing is that we have a strict inequality in \( 0 < R < \infty \) and, because of the definition of the ratio test, we have no idea what happens when \( \left| x-a \right| = R \). The series could converge or diverge. The ratio test doesn't give us a clue on what happens in that case. That's where we need to find the interval of convergence, which we discuss next.

Interval of Convergence

We use the radius of convergence, \(R\), to calculate the interval of convergence as follows
\( \begin{array}{rcccl} & & \left| x-a \right| & < & R \\ -R & < & x-a & < & R \\ -R + a & < & x & < & R + a \end{array} \)
So now we have an open interval \( (-R+a, R+a) \) in which the series converges. Now we need to look at the endpoints to determine what goes on. To do that we substitute each endpoint individually for \(x\) into the series and then use the other series test to determine convergence or divergence.

Notice when we substitute \(x=-R+a\) into the \((x-a)^n\) term, we end up with \((-R)^n\) which can be simplified to \((-1)^n R^n\). Now we have an alternating series. So often, the alternating series test can be used to determine convergence or divergence. The point is that, using other tests, we need to definitively determine convergence or divergence at each endpoint. The result will be an open interval, a half-open interval or a closed interval. We call this interval, the interval of convergence.

Notice the difference between the terms radius of convergence and interval of convergence. The radius of convergence gives information about the open interval but says nothing about the endpoints. The interval of convergence includes the radius of convergence AND information about convergence or divergence of the endpoints.

See the radius and interval of convergence page page for more detail, videos and practice problems.

List of Common Power Series

This panel contains a list of common power series. These are often used to build other power series. See the practice problems below for examples. You can find more discussion on the Taylor Series page.

function

power series

convergence interval

\(\displaystyle{ \frac{1}{1-x} }\)

\(\displaystyle{ \sum_ {n=0}^{\infty}{x^n} }\)

\( -1 < x < 1 \)

power series

\(e^x\)

\(\displaystyle{ \sum_{n=0}^{x^n}{\frac{x^n}{n!}} }\)

\(\displaystyle{ 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots }\)

\(-\infty < x < \infty\)

Maclaurin series

\(\ln(x)\)

\(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{(-1)^{n+1} (x-1)^n}{n} } }\)

\(\displaystyle{ (x-1) - \frac{(x-1)^2}{2} + \frac{(x-1)^3}{3} - \frac{(x-1)^4}{4} + \cdots }\)

Taylor series about \( x = 1 \)

\(\sin(x)\)

\(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{(-1)^n x^{2n+1}}{(2n+1)!} } }\)

\(\displaystyle{ x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots }\)

\(-\infty < x < \infty\)

Maclaurin series

\(\cos(x)\)

\(\displaystyle{ \sum_{n=0}^{\infty}{ \frac{(-1)^n x^{2n}}{(2n)!} } }\)

\(\displaystyle{ 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots }\)

\(-\infty < x < \infty\)

Maclaurin series

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Practice Problems

Instructions - - Unless otherwise instructed,
- if you are given a power series, determine the function that the sum represents. Assume that the values of x are such that the series converges.
- if you are given a function, build the power series of the function at the given point (if no point is given, use \(x=0\)), and determine the radius of convergence.

Level A - Basic

Practice A01

\(\displaystyle{g(x)=\frac{1}{1-x^3}}\)

answer

solution

Practice A02

\(\displaystyle{h(x)=\frac{1}{x-6}}\)

answer

solution

Practice A03

\(\displaystyle{\frac{6x^2}{7+15x}}\)

answer

solution

Practice A04

\(\displaystyle{f(x)=\frac{x^5}{8+x^2}}\)

answer

solution

Practice A05

\(\displaystyle{\sum_{n=0}^{\infty}{\frac{x^{n+2}}{n!}}}\)

answer

solution

Practice A06

\(\displaystyle{\sum_{n=2}^{\infty}{x^n}}\)

answer

solution

Practice A07

\(\displaystyle{\sum_{n=0}^{\infty}{\left[\frac{x^n}{n!}+x^n\right]}}\)

answer

solution

Practice A08

\(\displaystyle{\sum_{n=-1}^{\infty}{x^{n+1}}}\)

answer

solution

Practice A09

\(g(x)=(x+1)e^x\)

answer

solution

Practice A10

\(\displaystyle{\frac{1}{1+x^2}}\)

answer

solution

Practice A11

\(\displaystyle{f(x)=x^2\sin(x^3)}\)

answer

solution

Practice A12

\(\displaystyle{\frac{1}{1+x}}\)

answer

solution

Practice A13

\(\displaystyle{\frac{1}{1+9x^2}}\)

answer

solution

Practice A14

\(\displaystyle{\frac{x}{4x+1}}\)

answer

solution

Practice A15

\(\displaystyle{\frac{x}{9+x^2}}\)

answer

solution

Practice A16

\(\displaystyle{\sum_{n=1}^{\infty}{\frac{(-1)^{n+1}x^{2n}}{(2n-1)!}}}\)

answer

solution

Practice A17

\(x\cos(x)-\sin(x)\)

answer

solution

Practice A18

\(\displaystyle{f(x)=x^2e^{4x}}\)

answer

solution

Practice A19

For \(\displaystyle{f(x)=\sum_{n=0}^{\infty}{(5x-1)^n}}\), evaluate\(\int{f(x)~dx}\)

answer

solution

Practice A20

\(\displaystyle{f(x)=\sum_{n=0}^{\infty}{\frac{x^n}{n!}}}\), find \(\displaystyle{h(x)=x^3\int_{0}^{x}{f(t)~dt}}\)

answer

solution

Practice A21

\(\displaystyle{\frac{2}{x+4}}\)

answer

solution

Practice A22

\(\displaystyle{\cos(x^2)}\)

answer

solution


Level B - Intermediate

Practice B01

\(\ln(1+x)\)

answer

solution

Practice B02

\(\displaystyle{g(x)=\frac{1}{(x-1)^2}}\)

answer

solution

Practice B03

\(\displaystyle{\ln\left(\frac{1-x}{1+x}\right)}\)

answer

solution

Practice B04

\(\displaystyle{\frac{x^2}{(1+x)^3}}\)

answer

solution

Practice B05

\(\displaystyle{f(x)=\cos^2(x)}\)

answer

solution

Practice B06

\(\displaystyle{\frac{x^2}{(1-2x)^2}}\)

answer

solution

Practice B07

\(\displaystyle{e^{-x^2}\cos(x)}\)

answer

solution

Practice B08

\(\displaystyle{\frac{x}{\sin(x)}}\)

answer

solution

Practice B09

evaluate \(\displaystyle{\int{\frac{e^{x^2}}{x}dx}}\) as an infinite series

answer

solution

Practice B10

\(\displaystyle{\frac{6}{2x+1},~x=1}\)

answer

solution

Practice B11

\(\displaystyle{f(x)=\frac{x+3}{1-x^2}}\)

answer

solution

Practice B12

\(\displaystyle{f(x)=e^x\sin(x)}\)

answer

solution

Practice B13

\(\displaystyle{f(x)=\tan(x)}\)

answer

solution

Practice B14

use the binomial series to expand \(\displaystyle{\frac{1}{\sqrt[5]{32-x}}}\) as a power series

answer

solution


Level C - Advanced

Practice C01

\(\displaystyle{g(x)=\tan^{-1}(x)}\)

answer

solution

Practice C02

\(\displaystyle{\sum_{n=1}^{\infty}{\frac{n}{5^n}}}\)

answer

solution

Practice C03

\(\displaystyle{f(x)=\arctan(2x)}\)

answer

solution

Practice C04

use a power series to approximate \(\displaystyle{\int_{0}^{0.5}{\frac{\sin(x)}{x}dx}}\) with error \(< 0.001\)

answer

solution

Practice C05

find the maclaurin series using the binomial series for \(\arcsin(x)\)

answer

solution

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