Proving divergence of series

Question

I am attempting to establish divergence of the following series at $x=1/e$ $$\sum_{n=1}^\infty x^{1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}}$$ Most of my ideas have revolved around the comparison test, but I haven't been able to find a divergent series which is less than my original series. Any help would be appreciated.

Answer 1

Render

$(1/1)+(1/2)+(1/3)+...+(1/n)<1+\int_1^n(1/t)dt=1+\ln(n)$

due to $f(t)=1/t$ being monotonically decreasing for positive $t$. Then

$(1/e)^{(1/1)+(1/2)+(1/3)+...+(1/n)}>(1/e)^{1+\ln(n)}=(1/(en))$

and your series compares with the harmonic series.