Derivatives and Integrals of Power Series

As long as we are strictly inside the interval of convergence, we can take derivatives and integrals of power series allowing us to get new series. 

(Let $R$ be the radius of convergence of a series.  $x$ is strictly inside the interval of convergence of the series when $-R<x<R$, so $x$ is not equal to either of the two possible endpoints $R$ or $-R$ of the interval.) 

To see how this works with a series centered at the origin, first consider that for any constant $c_n$, $\frac d{dx}\left(c_nx^n\right)=nc_nx^{n-1}$.  Similarly, $\int c_nx^n\,dx=c_n\frac{x^{n+1}}{n+1} + C$.  Now consider the power series $\displaystyle\sum_{n=0}^\infty c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5+\cdots$.  When $x$ is strictly inside the interval of convergence of this series, we can differentiate and integrate term by term.  A beautiful fact is that the radius of convergence will not change when you differentiate or antidifferentiate.

To differentiate, we simply differentiate each term (not worrying that we have infinitely many terms) and then put the terms back into summation notation.  Notice that in the derivative series we must change our index to begin at $n=1$. (Why?)
$\begin{eqnarray} \frac{d}{dx}\left(\sum_{n=0}^\infty c_n \,x^n\right)&=&\frac{d}{dx}(c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5+\cdots)\\ &=&0+c_1+2c_2x+3c_3x^2+4c_4x^3+5c_5x^4+\cdots\\ &=& \sum_{n=1}^\infty n\,c_n \,x^{n-1} \end{eqnarray}$

Similarly, while we get an infinite integrand, we don't worry and just antidifferentiate each term, and then add a constant.  We do not need to change our index starting point here.  (Why not?)

$\begin{eqnarray} \int\left(\sum_{n=0}^\infty c_n \,x^n\right)\,dx&=&\int(c_0+c_1x+c_2x^2+c_3x^3+c_4x^4+c_5x^5+\cdots)\,dx\\ &=&c_0x+c_1\frac{x^2}{2}+c_2\frac{x^3}{3}+c_3\frac{x^4}{4}+c_4\frac{x^5}{5}+c_5\frac{x^6}{6}\cdots\\ &=& \left(\sum_{n=0}^\infty \,c_n \,\frac{x^{n+1}}{n+1}\right)+C \end{eqnarray}$

Succinctly, we get the following for power series centered at the origin:

The same holds for power series centered at $a$.