series stuff
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
$$\sum_{n=1}^{\infty} ( 1 + \frac{ 1 }{ n })^{n^2} $$
Using the DCT i can compare this to
$$\lim_{n \rightarrow \infty} (1 + \frac{ 1 }{ n })^n$$
but i'm not sure how to find the limit of this one ( i know it converges to e, but how so?)

Question

series stuff
Determine whether the series is absolutely convergent, conditionally convergent, or divergent.
$$\sum_{n=1}^{\infty} ( 1 +  \frac{ 1 }{ n })^{n^2} $$
Using the DCT i can compare this to
$$\lim_{n \rightarrow \infty} (1 + \frac{ 1 }{ n })^n$$
but i'm not sure how to find the limit of this one ( i know it converges to e, but how so?)

ganeshie8 · Answer

isn't it simply the definition of e ?

ganeshie8 · Answer

By definition : $$\lim_{n \rightarrow \infty} (1 + \frac{ x }{ n })^n=e^x$$

shamil98 · Answer

how do i prove that though?

ganeshie8 · Answer

you don't prove definitions

ganeshie8 · Answer

try this http://math.stackexchange.com/questions/358830/about-lim-left1-frac-xn-rightn

shamil98 · Answer

thanks

ganeshie8 · Answer

Np :) 
I think you're using "limit test", not DCT