Is this series convergent or divergent.?

Question

Is 1/sqrt(n) divergent?

If it is would that make 1/sqrt(2^n * n) divergent since it is greater than 1/sqrt(n)

Answer 1

Use the integral test:

int { 0 to infinity} 1 / sqrt(x) dx = int {0 to infinity} x^(-1/2) dx = 2 x^(1/2) {0 to infinity} which diverges, so the series diverges.

Answer 2

Do a comparison to the series 1/n, which is divergent. Term by term, n^(1/2) is larger than 1/n, so it also diverges.