The root test for convergence

When the root test indicates absolute convergence, divergence, or is inconclusive

The root test for convergence lets us determine the convergence or divergence of a series $a_n$ using the limit

$L=\lim_{n\to\infty}\sqrt[n]{|a_n|}$

The convergence or divergence of the series depends on the value of $L$ .

the series converges absolutely if $L<1$ .

the series diverges if $L>1$ or if $L$ is infinite.

the test is inconclusive if $L=1$ .

The root test is used most often when our series includes something raised to the $n$ th power.

Example

Use the root test to say whether the series converges or diverges.

$\sum^{\infty}_{n=1}\frac{6^n}{(n+2)^n}$

To use the root test, we need to solve for the limit

$L=\lim_{n\to\infty\sqrt[n]{|a_n|}}$

and then evaluate the value of $L$ .

$L=\lim_{n\to\infty}\sqrt[n]{\left|\frac{6^n}{(n+2)^n}\right|}$

We can drop the absolute value bars since all of our terms will be positive.

$L=\lim_{n\to\infty}\sqrt[n]{\frac{6^n}{(n+2)^n}}$

$L=\lim_{n\to\infty}\left[\frac{6^n}{(n+2)^n}\right]^{\frac{1}{n}}$

$L=\lim_{n\to\infty}\left[\left(\frac{6}{n+2}\right)^n\right]^{\frac{1}{n}}$

$L=\lim_{n\to\infty}\left(\frac{6}{n+2}\right)^\frac{n}{n}$

$L=\lim_{n\to\infty}\frac{6}{n+2}$

$L=\frac{6}{\infty+2}$

$L=\frac{6}{\infty}$

$L=0$

Since $L<1$ , we can say that the original series $a_n$ converges absolutely.