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?)

Answer 1

isn't it simply the definition of e ?

Answer 2

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

Answer 3

how do i prove that though?

Answer 4

you don't prove definitions

Answer 5

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

Answer 6

thanks

Answer 7

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