Theorem 8.3.10: Differentiating and Integrating Power Series
Let
an (x - c)n
be a power series centered at 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.
an (x - c)n dx = an (x - c)n dx = 1/n+1 an (x - c)n+1 + const- The power series is differentiable and can be differentiated term-by-term for all |x - c| < r, i.e.
an (x - c)n = an (x - c)n = n an (x - c)n-1