Please don't solve.
$$\sum_{n=1}^{\infty} \frac{(2n+1)^{n}}{n^{2n}}$$
I was wondering if you can give me hint on what method i should use to test for convergence.
I tried diving by n...but I feel like I'm not doing the division right.
$$\frac{{(2+\frac{1}{n})}^{n}}{1}$$

Question

Please don't solve.
$$\sum_{n=1}^{\infty} \frac{(2n+1)^{n}}{n^{2n}}$$
I was wondering if you can give me hint on what method i should use to test for convergence.
I tried diving by n...but I feel like I'm not doing the division right.
$$\frac{{(2+\frac{1}{n})}^{n}}{1}$$

zarkon · Answer

root test

anonymous · Answer

just curious, how did you come to that conclusion?

anonymous · Answer

i cannot speak for zarkon, but i would say
because everything is being raised to a power of $n$ and so it is natural to take the nth root 
if you had factorials the ratio test would come to mind because terms tend to cancel

anonymous · Answer

$$a_{n} = \left( \frac{2n+1}{n^{2}} \right)^{n}$$

$$\frac{2}{n} =0$$
convergent?

anonymous · Answer

lim n-> $$\infty$$

anonymous · Answer

is this a different question?

anonymous · Answer

no the same...what did i do wrong?

anonymous · Answer

i see..... im supposed to see if L < or > or = 1   right?

anonymous · Answer

oh i see i wasn't paying attention 
nth root is 
$$\frac{2n+1}{n^2}$$

anonymous · Answer

take the limit as $n\to \infty$ and get 0, so you are in good shape (i.e. converges)