It has been more than 7 days I have been trying to prove this following result using Harmonic Numbers.
Let me add this:
solution doesn't answer my question as I'm looking for the proof-based using Harmonic Numbers. If possible, I would like to avoid integration for the series so that I at least could get the intuitional background.
My attempts:
I could prove
The LHS: $ \left(1-\frac13+\frac15-\frac17+\cdots\right)^2= \lim_{x\to 1}(\tan^{-1}x)^2 $ which because of the this square of the series I was looking for a non-squared series which later
$$ \tan^{-1}\frac1x = \sum_{k\ge0}\frac{(-1)^k}{(2k+1)x^{2k}} $$
Which later for $ (\tan^{-1}x)^2 =(\sum_{k\ge0}\frac{(-1)^k}{(2k+1)x^{2k}} )^2 $
Or $$ (\tan^{-1}x)^2 = \sum_{q\ge0}\sum_{p\ge0} \frac {(-1)^{p+q}}{(2p+1)(2q+1)}x^{2{(p+q+1)}} $$
In the end I could find this beautiful formula (For me at least):
$$ \color{blue}{(\tan^{-1}(x))^2 = \sum_{k =1}^{\infty}\frac {(-1)^{k-1}}{k}h_kx^{2k} } $$
$$ \text{Where } h_k = \sum_{i =1}^{k}\frac 1{2i-1} = H_{2k} - \frac12H_k $$
Which later using:
$$ \sum_{k =1}^x a_kb_k = S_xb_x - \sum_{k =1}^{x-1}S_k(b_{k+1} - b_k) $$
$$ \sum_{k =1}^\infty\frac{(-1)^{k-1}(H_{2k} - \frac12H_{k})}{k} = \lim_{x\to\infty}\left(H'_x(H_{2x} -\frac12H_x) - \sum_{k =1}^{x -1}\frac {H'_k}{2k+1}\right) $$
I tried to use a few other identities so that I at least could end up to $ \zeta(2), Li_n(x) $ but I'm stuck and don't know how to go forward.
Also:
$$ f(x) = \sum_{k =1}^{\infty}\frac {h_kx^{2k-1}}{2k-1} \text { , } |x|< 1 $$
$$ f\left(\frac x{2-x}\right) = \frac18\log^2(1-x) + \frac12Li_2(x) $$
Where, $ Li_n(1) = \zeta(n) $
I do not want to use facts like $ \frac π4,\frac{π^2}{6} $. I'm looking for the complete proof that bridges square of alternate odd harmonic number series with a sum of reciprocals of squared natural numbers.
Why did I add this problem to MSE?
I'm really not able to find time for solving maths and can't live stable without solving it at the same time. Even If I'll have time most of all my time goes into finding things that are already (invented) as I lack mathematical background. Most of all of the above formulas are from Ramanujan's lost notebook and most of all my time goes in shuffling pages... I hope MSE can help me live obsessed-free life... haha
unsolved