Question

How do you know when to use the Root Test for convergence of a series?

Answer 1

I would use Root Test when the terms of the series are in the form of some expression to the nth power; otherwise, I would try other tests first.

---

Example

Let us look at examine the convergence of the series:

`sum_{n=1}^infty({2n}/{5-3n})^n`

By Root Test,

`lim_{n to infty}root{n}{|({2n}/{5-3n})^n|}=lim_{n to infty}|{2n}/{5-3n}|`

by dividing the numerator and the denominator by `n`,

`=lim_{n to infty}|{2}/{5/n-3}|=|{2}/{0-3}|=2/3<1`

Hence, the series is absolutely convergent.

---

I hope that this was helpful.