How to prove that Lp norm exists?

Question

how do I show that

$$\sqrt[\leftroot{-1}\uproot{4}p]{\sum_{n=1}^{\infty} \frac{n^p} {2^n}}$$

is finite?

I believe this is a type of Lp norm?

Answer 1

Apply ratio test. $\frac {a_{n+1}} {a_n} \to \frac 1 2$.

Answer 2

The sum converges by the ratio test, or the root test.