Root Test

Theorem

Consider the Series ${\sum }_{n=1}^{\infty }{a}_n$. Let

$$\rho =\underset{n\to \infty }{\lim }\sqrt[n]{|a_n|}.$$

i. If $0\le \rho <1$, then the series converges absolutely.

ii. If $\rho >1$ or $\rho =\infty$, then the series diverges.

iii. If $\rho =1$, then the test does not provide any information.

The root test is useful for series whose terms involve exponentials. In particular, for a series whose terms $a_n$ satisfy $|a_n|=b^n$, then $\sqrt[n]{|a_n|}=b$, and we need only evaluate ${\lim }_{n\to \infty }{b}_n$.