# determine of series converges or diverges

• Mar 22nd 2009, 02:08 PM
twilightstr
determine of series converges or diverges
the sum (1+ 1/n)^2e^-n
diverges or converges?
• Mar 22nd 2009, 02:28 PM
NonCommAlg
Quote:

Originally Posted by twilightstr

the series $\displaystyle \color{red}\sum_{n=1}^{\infty} (1+ 1/n)^2e^{-n}$ diverges or converges?

it converges because: $\displaystyle (1 + 1/n)^2 e^{-n} \leq 4e^{-n}.$
• Mar 22nd 2009, 02:34 PM
o_O
Ok I can't really tell what the series is but I assumed it was : $\displaystyle \sum \left(1 + \frac{1}{n}\right)^{\displaystyle 2e^{-n}}$

Use the fact that: If $\displaystyle \lim_{n \to \infty} a_n \neq 0$, then $\displaystyle \sum a_n$ diverges.

So, raise $\displaystyle a_n$ to the power of $\displaystyle 1 = \tfrac{n}{n}$ to get:

$\displaystyle \lim_{n \to \infty} \left[\left(1 + \frac{1}{n}\right)^{n}\right]^{\frac{2}{ne^n}} = \lim_{n \to \infty}e^{\frac{2}{ne^n}}$

etc. etc.
• Mar 22nd 2009, 03:10 PM
twilightstr
the problem is the way noncommalg wrote it.