Help with sum of infinite series using the root test.

[iluvphysics]

Homework Statement

Does it converge or diverge?

Sum n=0 to infinity : (n/(n+1))^(n^2)

The attempt at a solution

I know I need to use the root test.
But what I get is ...

Limit to infinity : (n/(n+1))^n

It seems that the n/(n+1) would go to 1 because when you multiply the top and bottom by n^-1 to get 1/(1+1/n) the 1/n goes to zero and you have 1/1. 1^n is 1, and according to the root test, it is inconclusive. However, the series converges and apparently the limit goes to 1/e, but I'm not sure how.

Also, does the latex syntax not work in preview mode or something?

 

[vela]

To evaluate

$$\lim_{n\rightarrow\infty} (1+1/n)^n$$

try finding the limit of the log of the function.

 

[iluvphysics]

To evaluate

$$\lim_{n\rightarrow\infty} (1+1/n)^n$$

try finding the limit of the log of the function.

I have no idea how.

 

[HallsofIvy]

Are you saying that you do not know how to find a logarithm??

If $y= (1+ 1/n)^n$, then $log(y)= n log(1+ 1/n)$. What is the limit of that as n goes to infinity? Perhaps L'Hopital's rule or writing log(1+ 1/n) as a Taylor's series will help.

 

[Sweet_GirL]

Homework Statement

Does it converge or diverge?

Sum n=0 to infinity : (n/(n+1))^(n^2)

The attempt at a solution

I know I need to use the root test.
But what I get is ...

Limit to infinity : (n/(n+1))^n

It seems that the n/(n+1) would go to 1 because when you multiply the top and bottom by n^-1 to get 1/(1+1/n) the 1/n goes to zero and you have 1/1. 1^n is 1, and according to the root test, it is inconclusive. However, the series converges and apparently the limit goes to 1/e, but I'm not sure how.

Also, does the latex syntax not work in preview mode or something?

$$\lim_{n\to\infty} \left( \frac{n}{n+1} \right)^n = \lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^{-n} = ?$$

Recall:
$$\lim_{n\to\infty} \left( 1 + \frac{a}{n^k} \right)^{bn^k} = e^{ab}$$