Power Series Math Example 3

Follow the full solution, then compare it with the other examples linked below.

easy

Find the radius of convergence of

\displaystyle\sum_{n=0}^{\infty} \frac{x^n}{n!}

Answer

R = \infty

(converges for all

x

Factorial in the denominator ensures convergence everywhere. This is the

e^x

series.

About Power Series

An infinite series of the form

\sum_{n=0}^{\infty} a_n(x-c)^n = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots

where

c

is the center and

a_n

are the coefficients. A power series defines a function of

x

wherever it converges.

Find the radius of convergence of [formula].

Find the interval of convergence of [formula].

Differentiate [formula] term by term to find a new series identity.