Root Test for for Convergence of an Infinite Series
Key Questions

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}/{53n})^n# By Root Test,
#lim_{n to infty}root{n}{({2n}/{53n})^n}=lim_{n to infty}{2n}/{53n}# by dividing the numerator and the denominator by
#n# ,#=lim_{n to infty}{2}/{5/n3}={2}/{03}=2/3<1# Hence, the series is absolutely convergent.
I hope that this was helpful.

Let
#a_n=({n^2+1]/{2n^2+1})^n# .By Root Test,
#lim_{n to infty}root[n]{a_n}=lim_{n to infty}root[n]{({n^2+1}/{2n^2+1})^n}# by cancelling out the nthroot and the nthpower,
#=lim_{n to infty}{n^2+1}/{2n^2+1}# (Note: the absolute value is not necessary since it is already positive.)
by dividing by
#n^2# ,#=lim_{n to infty}{1+1/n^2}/{2+1/n^2}={1+0}/{2+0}=1/2<1# Hence, the series converges.
I hope that this was helpful.

Root Test
If
#lim_{n to infty}root[n]{a_n}<1# , then#sum_{n=1}^inftya_n# converges.
If#lim_{n to infty}root[n]{a_n}>1# , then#sum_{n=1}^inftya_n# diverges.
If#lim_{n to infty}root[n]{a_n}=1# , then it is inconclusive.
I hope that this was helpful.
Questions
Tests of Convergence / Divergence

Geometric Series

Nth Term Test for Divergence of an Infinite Series

Direct Comparison Test for Convergence of an Infinite Series

Ratio Test for Convergence of an Infinite Series

Integral Test for Convergence of an Infinite Series

Limit Comparison Test for Convergence of an Infinite Series

Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series

Infinite Sequences

Root Test for for Convergence of an Infinite Series

Infinite Series

Strategies to Test an Infinite Series for Convergence

Harmonic Series

Indeterminate Forms and de L'hospital's Rule

Partial Sums of Infinite Series