TITLE: Show that $\frac{1}{2}(0.5)+\frac{(1+\frac{1}{2})}{3}(0.5)^2+\frac{(1+\frac{1}{2}+\frac{1}{3})}{4}(0.5)^3+\cdots=(\log 2)^2$
QUESTION [5 upvotes]: Show that $$\frac{1}{2}(0.5)+\frac{(1+\frac{1}{2})}{3}(0.5)^2+\frac{(1+\frac{1}{2}+\frac{1}{3})}{4}(0.5)^3+\cdots=(\log 2)^2$$
I tried to prove it by using the Taylor series, But I've found it not useful as shown below

any help, thanks

REPLY [2 votes]: $$\begin{align}
\sum_{n=2}^{\infty} \frac{H_{n-1}}{n 2^{n-1}}=\sum_{n=2}^{\infty}\sum_{m=1}^{n-1} \frac1{n 2^{n-1} m}
\\=\sum_{m=1}^{\infty}\sum_{n=m+1}^{\infty}\frac1{n 2^{n-1} m}
\\=2\sum_{n,m=1}^{\infty}\frac1{m(n+m)2^n\,2^m}
\\=2\sum_{n,m=1}^{\infty}\frac1{n(n+m)2^n\,2^m}
\\=\sum_{n,m=1}^{\infty}\frac1{n 2^n}\frac1{m 2^m}=\log^2 2
\end{align}$$