## Theorem 8.3.10: Differentiating and Integrating Power Series

Let

*a*be a power series centered at_{n}(x - c)^{n}*c*with radius of convergence*r > 0*. Then:

- The power series represents a continuous function for
|x-c| < r- The power series is integrable and can be integrated term-by-term for all
|x - c| < r, i.e.a_{n}(x - c)^{n}dx = a_{n}(x - c)^{n}dx =^{1}/_{n+1}a_{n}(x - c)^{n+1}+ const- The power series is differentiable and can be differentiated term-by-term for all
|x - c| < r, i.e.a_{n}(x - c)^{n}= a_{n}(x - c)^{n}= n a_{n}(x - c)^{n-1}