Question

determine the limit of the sequence if convergent.
An= (1+(2/n))^n

Answer 1

First establish that \(a_n\) is a convergent sequence. Show that it's monotonic and bounded, then you can go ahead with finding the limit.

Answer 2

SaG, I know thats the general form to go about, and im sure that ln is used, however i have zero to none on ideas to solve this problem. If you have specifics that would be sith tons of awesomeness

Answer 3

To show it's monotonic: Consider the function
\[f(x)=\left(1+\frac{2}{x}\right)^x\]
You can check for monotonicity by examining the derivative:
\[\begin{align*}\ln f(x)&=x\ln\left(1+\frac{2}{x}\right)\\\\\\
\frac{1}{f(x)}f'(x)&=\ln\left(1+\frac{2}{x}\right)+x\left(\frac{-\dfrac{2}{x^2}}{1+\dfrac{2}{x}}\right)\\\\\\
\frac{1}{f(x)}f'(x)&=\ln\left(1+\frac{2}{x}\right)-\frac{2}{x+2}\\
f'(x)&=\left(1+\frac{2}{x}\right)^x\left(\ln\left(1+\frac{2}{x}\right)-\frac{2}{x+2}\right)
\end{align*}\]
Find the critical points, etc. Figure out for which \(x\) the function is either strictly decreasing or increasing.

Answer 4

The limit is \[ e^2\]

Answer 5

Use L'Hospital Rule

Answer 6

on log(f(x))

Answer 7

http://www.wolframalpha.com/input/?i=limit++%281%2B%282%2Fn%29%29^n+when+n+goes+to+infinity

Answer 8

To show it's bounded: Find an upper bound (if the sequence is increasing) or a lower bound (if the sequence is decreasing).

Recalling the limit
\[\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e\]
might help.

You could also try listing the first few terms to see if you can extrapolate what the limit might be.