Editing Power series (section)

=== Differentiation and integration===
Once a function <math>f(x)</math> is given as a power series as above, it is [[derivative|differentiable]] on the [[interior (topology)|interior]] of the domain of convergence. It can be [[derivative|differentiated]] and [[integral|integrated]] by treating every term separately since both differentiation and integration are linear transformations of functions:
<math display="block">\begin{align}
    f'(x) &= \sum_{n=1}^\infty a_n n (x - c)^{n-1} = \sum_{n=0}^\infty a_{n+1} (n + 1) (x - c)^n, \\
  \int f(x)\,dx &= \sum_{n=0}^\infty \frac{a_n (x - c)^{n+1}}{n + 1} + k = \sum_{n=1}^\infty \frac{a_{n-1} (x - c)^n}{n} + k.
\end{align}</math>

Both of these series have the same radius of convergence as the original series.