Why does this series diverge using Ratio Test?

Question

So I was working on this problem

$$\sum\limits_{n=2}^\infty \frac{2^n}{n^{100}}$$

After all of my work performing the ratio test, I ended up with a $$\frac{2}{n+1} = 0$$

Therefore, I assumed it absolutely converges, but turns out that it diverges. May I please get an explanation why?

Answer 1

I think you computed the ratio incorrectly. It should be $$\frac{2^{n+1}}{(n+1)^{100}} \cdot \frac{n^{100}}{2^n} = 2 \left(\frac{n}{n+1}\right)^{100}$$

Answer 2

$$\frac{2^{n+1}}{(n+1)^{100}}\frac{n^{100}}{2^n}=\frac{2}{(1+\frac{1}{n})^{100}}$$

Answer 3

A simple remark: you don't even need the ratio test, as this series diverges trivially: its general terù $\frac{2^n}{n^{100}}$ does not tend to $0$ at $\infty$.