The Ratio and Root Tests

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So which test should we use? Whichever is easier!

If $a_n$ involves factorials, then it's probably easier to compute $\displaystyle\frac{a_{n+1}}{a_n}$ than $\lvert a_n\rvert^{1/n}$ , so the ratio test is probably best.

If $a_n$ involves $n^\text{th}$ powers, then computing $\lvert a_n\rvert^{1/n}$ is easy, and the root test is probably best.

The two tests give almost exactly the same information. If $R<1$ , then $\rho <1$ , if $R=1$ then $\rho=1$ , and if $R>1$ then $\rho>1$ . There are a few cases where the limit $\rho$ exists but $R$ doesn't, but these are rare.

Examples: The video below shows

how to use the root test to deduce the convergence of the series $\sum_{n=1}^\infty\frac{(3n+137)^n}{(4n+1)^n}$
how to use the ratio test to deduce the convergence of the series $\sum_{n=1}^\infty\frac{n!}{n^n}$
and how the series $\sum_{n=1}^\infty\frac{x^n}{n!}$ converges by the ratio test no matter what $x$ we choose.