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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

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