Calculation suggests the following identity: $$ \lim_{n\to \infty}\sum_{k=1}^{n}\frac{(-1)^k}{k}\sum_{j=1}^k\frac{1}{2j-1}=\frac{1-\sqrt{5}}{2}. $$

I have verified this identity for $n$ up to $5000$ via Maple and find that the left-hand side approaches $\frac{1-\sqrt{5}}{2}$. However, this double summation has slow rate of convergence and I am unsure it is true.

So I want to ask if it is true. If so, how to prove it?

(I used Euler-summation to order 0.58)With W/A you'd found the correct value easily... $\endgroup$2more comments