how to find the series $x + x^{1 + \frac{1}{2}} + x^{1 + \frac{1}{2}+ \frac{1}{3}} +...$ is convergent.
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 $\log x < -1\iff x < \frac 1e$.