Convergence of $ \sum\limits_{n=1}^{\infty}x^{\sqrt{n}}$ I am finding the positive values of $x$ for which the following series is convergent $$ \sum_{n=1}^{\infty}x^{\sqrt{n}}$$  It is sure that it is not convergent for $x\geq1$ as $n$-th term will not tend to zero. Now $x\in[0,1)$ how to check its convergence? Please help me to solve it. Thanks.  

Without the exponential:
If $0\le x <1$, we have:
\begin{align*}
\sum_{n\ge1}x^{\sqrt n}&\le\sum_{n\ge1}x^{\lfloor\sqrt n\rfloor}=3x+5x^2+7x^3+9x^4+\dotsm \\
&\le 2+4x+6x^2+8x^3+10x^4+\dotsm=2(1+2x+3x^2+4x^3+5x^4+\dotsm)\\
&= 2\biggl(\frac1{1-x}\biggr)'=\frac2{(1-x)^2}
\end{align*}
hence the series converges.
