Convergence of a series ${}\qquad{}$

Question

Does this series converge? $$\sum_{x=2}^n \left(\frac{1}{x}\right)^{\left(\frac{1}{x}\right)}$$ I tried hard but stil had problems...

Could someone help me?

Answer 1

Write

$$(1/x)^{1/x} = e^{\frac{\ln(1/x)}{x}}.$$

Now use L'Hospitals rule (or somethine else) to show that your summands converge to $1$ for $x \rightarrow \infty$.

Therefore, the series can not converge.

BTW: I assumed you meant $\sum_{x=2}^{\infty} (1/x)^{1/x}$.

Answer 2

Hint: $x < x^\alpha$ for $0 < x,\alpha < 1$