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## Derivatives and Integrals of Power Series As long as we are (Let $R$ be the radius of convergence of a series. To see how this works 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 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. $\begin{eqnarray} Succinctly, we get the following for
The same holds for
The new series must be tested at the endpoints
to determine their convergence. |