Question

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

Answer 1

HI!!

Answer 2

what is your definition of \(e^x\)?

Answer 3

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

Answer 4

in which case replacing \(x\) by \(2\) gives \(e^2\) right of the bat

Answer 5

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 \(e^2\)

Answer 6

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

Answer 7

Thank you!