Theorem 8.3.10: Differentiating and Integrating Power Series

Let a_n (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.

  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

Context