can someone help me understand why this is equal to e^2...Limit as n- infinity of [1+(2/n)]^n

Question

can someone help me understand why this is equal to e^2...Limit as n-> infinity of [1+(2/n)]^n

misty1212 · Answer

HI!!

misty1212 · Answer

what is your definition of $e^x$?

misty1212 · Answer

one is 
$$\lim_{n\to \infty}\left(1+\frac{x}{n}\right)^n$$

misty1212 · Answer

in which case replacing $x$ by $2$ gives $e^2$ right of the bat

misty1212 · Answer

elsewise you have to take the log and use l'hopitals rule for the resulting limit
you will get 2
so final answer will be $e^2$

irishboy123 · Answer

start with the original identity (from Bernoulli compounding formula, the origin of e)....

$$\lim_{n \rightarrow \infty} ( 1 + \frac{1}{n})^n = e$$

then play with some substitution, eg let k = 2*n or n/2 and do the limit for k -> infinity

kl0723 · Answer

Thank you!