Convergence/dicergence

Question

My series has a general term $\frac{(1+\frac{1}{n})^{n^2}}{e^n}$.

I found that the Root test is inconclusive here. Wolfram says to use "limit test". Is that the limit comparison test? Which series can I compare this one to? I know it should diverge.

Answer 1

Let $u_n$ be the general term.

$$\ln(u_n)=n^2\ln(1+\frac 1n)-n$$

$$\ln(1+\frac 1n)=\frac 1n -\frac{1}{2n^2}+\frac{1}{n^2}\epsilon(n)$$

$$\ln(u_n)=-\frac 12+\epsilon(n)$$

thus $$\lim_{n\to+\infty}u_n=\frac{1}{\sqrt{e}}\ne0$$

the series diverges by the limit test.