# What's the difference between radius of convergence and interval of convergence?

May 31, 2018

The radius of convergence is usually required to find the interval of convergence. While the radius gives us the number of values where the series converges, the interval gives us the exact values of where the series converges and doesn't. Take the following example.

sum_(n = 1)^oo(2^n (x+ 2)^n)/((n + 2)!)

We use the ratio test to find our radius of convergence. We know that the ratio states that if $\lim \left(n \to \infty\right) {a}_{n + 1} / {a}_{n} < 1$, the series converges. We want to find the values of $x$ where the series converges. Thus

lim_(n-> oo) ((2^(n + 1)(x + 2)^(n + 1))/((n + 1 + 2)!))/((2^n(x +2)^n)/((n + 2)!)) < 1

$2 \left(x + 2\right) {\lim}_{n \to \infty} \frac{1}{n + 3} < 1$

$0 < 1$

Since this is true for all values of $x$, the radius of convergence is $\infty$ and therefore the interval is $\left(- \infty , \infty\right)$.

Hopefully this helps!