how to find the series $x + x^{1 + \frac{1}{2}} + x^{1 + \frac{1}{2}+ \frac{1}{3}} +...$ is convergent. The series $x + x^{1 + \frac{1}{2}} + x^{1 + \frac{1}{2}+ \frac{1}{3}} +...$
is convergent if 
(A) $x>e$ 
(B) $x<e $
(C) $x<1/e$ 
(D) $x>1/e$
I think the answer is C, but I could not determine the condition... how to solve it. Plz help.

We have
\begin{align*}
  x^{\sum_{i=1}^n \frac 1i} &\le x^{\log n + 1}\\
   &= x \cdot \exp(\log n\cdot \log x)\\
   &= x \cdot n^{\log x}
\end{align*}
and
\begin{align*}
  x^{\sum_{i=1}^n \frac 1i} &\ge x^{\log n}\\
   &= \exp(\log n\cdot \log x)\\
   &= n^{\log x}
\end{align*}
and hence
\[ \sum_{n=1}^\infty x^{\sum_{i=1}^n \frac 1i} < \infty \iff \sum_{n=1}^\infty n^{\log x} < \infty \]
which is true exactly for $\log x < -1\iff x < \frac 1e$.
