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Are there any results/papers/books for series of the type $$\sum_{n=1}^{\infty}\frac{x^n}{n^p},\quad p \in \mathbb{R}\quad\text{?}$$ There are the special cases $p = 0$, $p=1$, which are well known. I would be interested especially in the cases $0 < p < 1$, in particular $p = 1/2$.

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The polylogarithm $Li(1/2,x)=\sum_{n=1}^\infty \frac{x^n}{n^{1/2}}$ has a radius of convergence of $\vert x\vert<1, x\neq-1$ following from the geometric series convergence criteria and the fact that $Li(1/2,1)=\zeta(1/2)$ which is divergent. Interestingly, for $Li(1/2,-1)$, the sum is equal to $-\eta(1/2)$ which is convergent.

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  • $\begingroup$ $\plog_{1/2}(1)\ne\zeta(1/2)$. $\zeta(1/2)$ is finite, and $\plog_{1/2}(1)$ doesn't exist. $\endgroup$ – Simply Beautiful Art Oct 29 '17 at 23:15
  • $\begingroup$ How is $Li(1/2,1)\neq\zeta(1/2)$? And using the traditional zeta definition, doesn't $\zeta(s)$ only converge for $\Re(s)>1$? $\endgroup$ – aleden Oct 29 '17 at 23:19
  • $\begingroup$ $\zeta(s)$ usually refers to the analytic continuation of $\sum_{n\ge1}\frac1{n^s}$, which is defined everywhere except $c=1$. $\endgroup$ – Simply Beautiful Art Oct 30 '17 at 0:20
  • $\begingroup$ Also $\text{Li}_s(z)$ is entire in $s$ and continuous in $z, |z| \le 1, z \ne 1$ $\endgroup$ – reuns Oct 30 '17 at 8:34

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