# Math Help - Series (ratio and root test)

1. ## Series (ratio and root test)

(n^sqrt(2) ) / 2^n

I want to determine if this series will converge or diverge. When I use the root test I get sqrt(2)/2 which is less than 1 so it will converge. The book used the ratio test. Could someone please show me how they used the ratio test to say it converges. I got lim n->infinity n/2 which is infinity so it would diverge.

2. You want to see if $\lim_{n \to \infty} \mid \frac{a_{n+1}}{a_n} \mid < 1$.

$\frac{(n+1)^{\sqrt{2}}}{2^{n+1}} \cdot \frac{2^{n}}{n^{\sqrt{2}}}$
$= \left ( \frac{n+1}{n} \right )^{\sqrt{2}} \cdot \frac{1}{2} \to \frac{1}{2}$ as $n \to \infty$