Working with some integrals I stumbled upon the following slowly converging series:
$$ S = \sum_{n = 0}^{\infty}\left(-1\right)^{n} \left[n + \frac{3}{2} + \left(n + 1\right)\left(n + 2\right) \log\left(1 - \frac{1}{n + 2}\right)\right] $$
I have reasons to suspect that the series has a closed form:
$$S=\frac{1}{2}-\frac{7 \zeta(3)}{2 \pi^2}=0.073721601182494209 \ldots$$
The actual proof eludes me so far.
Can you prove or disprove this conjecture?
Writing the logarithm as a series we have:
\begin{align} &\left(n + 2\right) \log\left(1 - \frac{1}{n+2}\right) = -\sum_{k = 1}^{\infty} \frac{1}{k\left(n + 2\right)^{k - 1}} \\ = &\ -1-\sum_{k=1}^\infty \frac{1}{(k+1) (n+2)^k} \end{align} Which turns the series into:
$$S=\sum_{n=0}^\infty (-1)^n \left(\frac{1}{2}-(n+1) \sum_{k=1}^\infty \frac{1}{(k+1) (n+2)^k} \right)$$
I can provide the way I came to this expression, but it's very long and complicated, as usual. I'd like some clear proof, if possible.
To get the sense of how slowly the series converges, for $20000$ terms the result agrees with the stated closed form in $4$ first significant digits.
The integral from which this series was obtained is (again, conjectured):
$$\int_0^1 {_2 F_1} (1,-t;2-t;-1) dt = \frac{7 \zeta(3)}{\pi^2}+\frac{1}{2}$$
I don't think it's very useful, except for numerical confirmation.