Power Series

Term by Term Operations on Series

f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n}

has radius of convergence

M \in ℝ^{+}

, then

f

is differentiable on

(- M, M)

and the following are true:

$\sum_{n = 1}^{\infty} n a_{n} x^{n - 1}$ has radius of convergence $M$ such that for any $x \in (- M, M)$ we have $f^{'} (x) = \sum_{n = 1}^{\infty} n a_{n} x^{n - 1}$
$\sum_{n = 0}^{\infty} \frac{a_{n}}{n + 1} x^{n + 1}$ has radius of convergence $M$ such that for any $x \in (- M, M)$ we have: $\int_{0}^{x} f (t) d t = \sum_{n = 0}^{\infty} \frac{a_{n}}{n + 1} x^{n + 1}$

Hadamard

Given a power series

\sum_{n = 0}^{\infty} a_{n} x^{n}

then either there exists a set

C

such that the series converges for any

x \in C

and diverges otherwise.

C

is either of the form

[- M, M]

ℝ

itself, such that

C = {\begin{matrix} ℝ & if α = 0 \\ \emptyset & if α = + \infty \\ [- \frac{1}{α}, \frac{1}{α}] & if α \in ℝ^{+} \end{matrix}