Question

How do you use the Root Test on the series `sum_(n=1)^oo((n^2+1)/(2n^2+1))^(n)` ?

Answer 1

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 nth-root and the nth-power,

`=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.