# Limits and sequences

• Sep 16th 2010, 06:48 PM
acevipa
Limits and sequences
Is this a standard result?

$\displaystyle \displaystyle\lim_{n\to\infty}a_n^{n}=\left( \lim_{n\to\infty}a_n\right)^n$

Because, if I'm given this question:

Find

$\displaystyle \displaystyle\lim_{n\to\infty}\left(1+\frac{1}{n}\ right)^{4n}$

It would equal to

$\displaystyle \displaystyle\lim_{n\to\infty}\left[\left(1+\frac{1}{n}\right)^n\left(1+\frac{1}{n}\ri ght)^n\left(1+\frac{1}{n}\right)^n\left(1+\frac{1} {n}\right)^n\right]$

$\displaystyle \displaystyle\left[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n\righ t]^4$
• Sep 16th 2010, 07:51 PM
Prove It
I believe it is correct, because the limit of a product is the same as the product of the limits, and exponentiation is "repeated multiplication".
• Sep 17th 2010, 01:51 AM
HallsofIvy
But the problem is NOT $\displaystyle \lim_{n\to\infty}\left(a_n\right)^n$, it is $\displaystyle \lim_{n\to\infty}\left(a_n\right)^4$ and that definitely is $\displaystyle \left(\lim_{n\to\infty} a_n\right)^4$.