Does this sum have an analytic form? Does this sum have an analytic form?
$$\sum_{n=1}^{\infty}\frac{a^{n}\left(-1\right)^{n}}{n\sqrt{n+1}}$$
Numerically, it appears to only converge on $-1 < a \le 1$

For analytic continuation purposes, we have (for $|a|\color{red}{\leqslant}1$) $$\sum_{n=1}^\infty\frac{(-a)^n}{n\sqrt{n+1}}=-a\int_0^\infty\frac{\operatorname{erf}\sqrt{t}}{a+e^t}\,dt,$$ with the RHS being analytic on $\color{blue}{a\in\mathbb{C}\setminus\mathbb{R}_{\leqslant-1}}$. The equality is obtained using $$\int_0^\infty e^{-st}\operatorname{erf}\sqrt{t}\,dt=\frac{1}{s\sqrt{s+1}}\quad(\Re s>0)$$ (this Laplace transform may be proved for $\Re s>1$ using the power series for $\operatorname{erf} z$ and termwise integration, and extended to $\Re s>0$ analytically; another, perhaps better, alternative is just integration by parts) and the geometric series $-a/(a+e^t)=\sum_{n=1}^\infty(-ae^{-t})^n$.
