# Sequence is Convergent or Divergent

$$(1+\frac{5}{n})^{2n}$$ I need to figure the limit if it converges or state if it goes to $$\pm\infty$$ or simply "divergent" if it diverges but not to infinity.

From what I can tell as n -> infinity $$\frac{5}{n} -> 0$$ so would it be the same as evaluating $$(1)^{2n}$$ which would just = 1 ?

or do i need to do some form of squeeze theorem like $$(\frac{5}{n})^{n} <(\frac{5}{n})^{2n} <(1+\frac{5}{n})^{2n}$$

## 2 Answers

We have

$$\left(1+\frac{5}{n}\right)^{2n}=\left[\left(1+\frac{1}{n/5}\right)^{n/5}\right]^{10}$$

and

$$\lim_{x\to\infty}\left(1+\frac{1}{x}\right)^x=e$$

The sequence $$\left(1+\frac an\right)^n$$ converges to $e^a$ as $n\to\infty$. So, your sequence tends to $e^{10}$.