Table of Contents

Differentiation and Integration of Power Series Examples 1
Recall from the Differentiation and Integration of Power Series page that:
 If $f(x) = \sum_{n=0}^{\infty} a_n (x  c)^n$ on the interval $(c  R, c + R)$ then $f$ is differentiable on $(c  R, c + R)$ and:
 If $f(x) = \sum_{n=0}^{\infty} a_n (x  c)^n$ on the interval $(c  R, c + R)$ then $f$ is integrable over any subinterval of $(c  R, c + R)$ and:
Furthermore, if the original series converges at either of the endpoints $c  R$ or $c + R$, then the differentiated series may not converge at these endpoints.
If the original series does not converge at either of the endpoints $c  R$ or $c + R$, then the integrated series may converge at either of these endpoints.
We will now look at some examples of differentiating and integrating power series.
Example 1
Find a power series representation for the function $f(x) = \ln \mid a  x\mid$ where $a > 0$, and determine the center of convergence $c$ and the radius of convergence $R$ for this series. Does the power series representation for $f(x) = \ln \mid a  x \mid$ converge at either of the endpoints $c  R$ or $c + R$? Determine the interval of convergence for this power series
We first note that $\ln \mid a  x \mid = \int \frac{1}{a  x} \: dx$.
We already have that:
(3)Therefore we have that:
(4)We now integrate the series above to get that:
(5)The center of convergence is $0$. Furthermore, we note that from the original series, $\frac{1}{1  x} = \sum_{n=0}^{\infty} x^n$ for $\mid x \mid < 1$. Therefore, $\frac{1}{1  \frac{x}{a}} = \sum_{n=0}^{\infty} \frac{x^n}{a^n}$ for $\biggr \rvert \frac{x}{a} \biggr \rvert < 1$ or equivalently, for $\mid x \mid < a$. Therefore the radius of convergence is $a$.
We will now test to see if our power series representation converges at either of the endpoints. First let's check $a$. We have that:
(6)The series above converges as the alternating harmonic series. Now let's check the endpoint $a$. We have that:
(7)The series above diverges as the harmonic series.
Thus we have that the interval of convergence is $[a, a)$.
Example 2
Determine a power series representation for $f(x) = \tan^{1} x$, and find the center or convergence and radius of convergence.
Recall that $\frac{d}{dx} \tan^{1} x = \frac{1}{1 + x^2}$ and so $\tan^{1} x = \int \frac{1}{1 + x^2} \: dx$. We have that:
(8)We now integrate this series to get that:
(9)This series has center of convergence $0$ and converges for $\mid x^2 \mid < 1$, or equivalently, $\mid x \mid < 1$.