stylistic:style_transfer
Differentiation and integration

Once a function f (x) {\displaystyle f(x)} is given as a power series as above, it is differentiable on the interior of the domain of convergence. It can be differentiated and integrated quite easily, by treating every term separately: f ′ (x) = ∑ n = 1 ∞ a n n (x − c) n − 1 = ∑ n = 0 ∞ a n + 1 (n + 1) (x − c) n, ∫ f (x) d x = ∑ n = 0 ∞ a n (x − c) n + 1 n + 1 + k = ∑ n = 1 ∞ a n − 1 (x − c) n n + k. {\displaystyle {\begin{aligned}f'(x)&=\sum _{n=1}^{\infty }a_{n}n(x-c)^{n-1}=\sum _{n=0}^{\infty }a_{n+1}(n+1)(x-c)^{n},\\\int f(x)\,dx&=\sum _{n=0}^{\infty }{\frac {a_{n}(x-c)^{n+1}}{n+1}}+k=\sum _{n=1}^{\infty }{\frac {a_{n-1}(x-c)^{n}}{n}}+k.\end{aligned}}}

Both of these series have the same radius of convergence as the original one.