Question: Use the Binomial Theorem and Squeeze Theorem to compute lim(1+(1/n^2))^n as n goes to infinity.
I used direct substitution using the binomial theorem, but can't seem to get very far. Any ideas?
Question: Use the Binomial Theorem and Squeeze Theorem to compute lim(1+(1/n^2))^n as n goes to infinity.
I used direct substitution using the binomial theorem, but can't seem to get very far. Any ideas?
I would rewrite the expression as follows:
$\displaystyle \left(1+\frac{1}{n^2} \right)^n=\left(\frac{n^2+1}{n^2} \right)^n=\frac{\left(n^2+1 \right)^n}{n^{2n}}$
Now, you need to find functions $\displaystyle f(n)$ and $\displaystyle g(n)$ such that:
$\displaystyle f(n)\le\left(n^2+1 \right)^n\le g(n)$
and where:
$\displaystyle \lim_{n\to\infty}\frac{f(n)}{n^{2n}}= \lim_{n\to\infty}\frac{g(n)}{n^{2n}}$
Can you find such functions?