A closed form for infinite series $ \sum _1 ^\infty \frac {1}{n^3} = \zeta (3) ?$ It is well known that $$ \sum _1 ^\infty \frac {1}{n^2} = \frac {\pi ^2}{6}$$ and $$ \sum _1 ^\infty \frac {1}{n^4} = \frac {\pi ^4}{90}$$ 
We also know that $$ \sum _1 ^\infty \frac {1}{n^3} $$
$$=1.202056903159594285399738161511449990764986292340498881.....$$
My question is:
Do we have a closed form for this  series  besides  $$ \sum _1 ^\infty \frac {1}{n^3}  =  \zeta (3) ?$$    
 A: Apéry's Constant
As mentioned in comments
$$
\sum_{n=1}^\infty\frac1{n^3}=\zeta(3)\tag1
$$
is also known as Apéry's Constant. There is no closed form in terms of a rational multiple of an integer power of $\pi$.
We also have
$$
\sum_{n=1}^\infty\frac1{(2n-1)^3}=\frac78\zeta(3)\tag2
$$
However, as shown in this answer, if we alternate the series in $(2)$, we get
$$
\sum_{n=1}^\infty\frac{(-1)^{n-1}}{(2n-1)^3}=\frac{\pi^3}{32}\tag3
$$
which is also known as $\beta(3)$, the Dirichlet beta function.

Computation of $\boldsymbol{\zeta(3)}$
The series in $(1)$ converges very slowly. To get the sum to about $10$ places, we would need to sum $100000$ terms.
If we use the Euler-Maclaurin Sum Formula, we get that
$$
\sum_{k=1}^n\frac1{k^3}=\small\zeta(3)-\frac1{2n^2}+\frac1{2n^3}-\frac1{4n^4}+\frac1{12n^6}-\frac1{12n^8}+\frac3{20n^{10}}-\frac5{12n^{12}}+O\!\left(\frac1{n^{14}}\right)\tag4
$$
Using $n=10$ in $(4)$, we get
$$
\sum_{k=1}^\infty\frac1{k^3}=1.2020569031596\tag5
$$
A: In addition to robojohn's answer, there are some formulas expressing $\zeta(3)$ (and other odd zeta values) in terms of powers of $\pi$, but these are not clsoed forms. The most well known ones are due to Plouffe and Borwein & Bradley:
$$
\begin{aligned}
\zeta(3)&=\frac{7\pi^3}{180}-2\sum_{n=1}^\infty \frac{1}{n^3(e^{2\pi n}-1)},\\
\sum_{n=1}^\infty \frac{1}{n^3\,\binom {2n}n} &= -\frac{4}{3}\,\zeta(3)+\frac{\pi\sqrt{3}}{2\cdot 3^2}\,\left(\zeta(2, \tfrac{1}{3})-\zeta(2,\tfrac{2}{3}) \right).
\end{aligned}
$$
Moreover, in this Math.SE post we have:
$$
\frac{3}{2}\,\zeta(3) = \frac{\pi^3}{24}\sqrt{2}-2\sum_{k=1}^\infty \frac{1}{k^3(e^{\pi k\sqrt{2}}-1)}-\sum_{k=1}^\infty\frac{1}{k^3(e^{2\pi k\sqrt{2}}-1)}.
$$
You can also check out this paper by Vepstas, which provides a nice generalization to some of these identities.
