Root Test

The Root Test is a tool we can use to indicate convergence of a series. The Root Test is particularly useful with n-th powers. If every single term in the original function is raised to the n-th power, then the root test is a great tool to determine convergence.

Given a series $\sum{a_n}$ , we can calculate

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

Then we can draw the following conclusions:

If $L<1$ , the series is absolutely convergent
If $L>1$ , the series is divergent
If $L=1$ , we can’t draw a conclusion from the root test, and therefore must use another test in order to determine the convergence of a series

To use the formula, draw big brackets around the original function, and raise the entire thing to the $\frac{1}{n}$ power. If every term in the original function is raised to the n-th power, then raising the whole function to the $\frac{1}{n}$ power will cancel out these powers.

Once you cancel the powers, evaluate the limit of the function as $n$ approaches $\infty$ . If your final answer is less than $1$ , then the series is absolutely convergent. If the final answer is greater than $1$ , the series is divergent. If your final answer is equal to $1$ , then you can’t draw a conclusion from the root test, so you’ll have to use another convergence test to determine the convergence or divergence of the series.

Root Test

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