We integrate power series term by term.

Integration of Power Series

Suppose that the power series converges for all in some open interval . Then, on this interval, the power series has an anti-derivative which can be obtained by integrating term-by-term:

In other words, the indefinite integral of a power series is computed term by term, as we would anti-differentiate a polynomial.

(problem 1) Find the anti-derivatives of the following power series. Also, compare the interval of convergence of the original series to its anti-derivative.