# What is $\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3} + 1}}$?

I am interested in the symbolic evaluation of infinite series over algebraic functions in terms of well-known special functions such as the Hurwitz zeta function. Often, infinite sums over algebraic expressions can be evaluated in a simple way in terms of well-known special functions. For example, a direct application of Euler's identity may be used to show that $$\sum_{n \in \mathbb{N}} \frac{1}{\sqrt{(n^2+1)(\sqrt{n^2+1}+n)}} = - \frac{i \left( \zeta\left( \frac{1}{2}, 1 - i \right) - \zeta\left(\frac{1}{2}, 1 + i \right) \right)}{\sqrt{2}}.$$ The above formula seems to suggest that there may be many classes of infinite series over algebraic expressions that could be easily evaluated in terms of well-established special functions.

Inspired in part by this question, together with this question and this question, as well as this question, I'm interested in the problem of evaluating $$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^{3} + 1}} = 2.29412...$$ in terms of "known" special functions, e.g., special functions implemented within Mathematica. This problem is slightly different from the related problems given in the above links, since I'm interested specifically in the use of special functions to compute the above sum symbolically.

More generally, what kinds of algorithms could be used to evaluate infinite sums over algebraic expressions symbolically in terms of special functions?

The given series can be written as $$\sum_{n\geq 1}\frac{1}{n^{3/2}}\cdot\frac{1}{\sqrt{1+\frac{1}{n^3}}}=\sum_{n\geq 1}\sum_{m\geq 0}\frac{(-1)^m\binom{2m}{m}}{4^m n^{3\left(m+\frac{1}{2}\right)}}=\sum_{m\geq 0}\frac{(-1)^m}{4^m}\binom{2m}{m}\zeta\left(3m+\frac{3}{2}\right)\tag{1}$$ or as $$\frac{1}{\sqrt{2}}+\sum_{m\geq 0}\underbrace{\frac{(-1)^m}{4^m}\binom{2m}{m}}_{\approx\frac{(-1)^m}{\sqrt{\pi m}}}\underbrace{\left[\zeta\left(3m+\tfrac{3}{2}\right)-1\right]}_{\approx\frac{C}{8^m}} \tag{2}$$ whose convergence speed is not so bad. $(1)$ also provides the integral representation $$\sum_{n\geq 1}\frac{1}{\sqrt{n^3+1}}=\frac{2}{\sqrt{\pi}}\int_{0}^{+\infty}\frac{\sqrt{z}}{e^z-1}\cdot \phantom{}_0 F_2\left(;\tfrac{5}{6},\tfrac{7}{6};-\tfrac{z^3}{27}\right)\,dz \tag{3}$$ allowing to approximate the ratio between the LHS and $\zeta\left(\tfrac{3}{2}\right)$ through the Cauchy-Schwarz inequality.