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

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Answer 1 · 2014-11-25T15:09:33

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.