I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question  number 769 which is about an infinite sum with logarithms. I didn't found a solution, so can someone please explain it to me.

https://books.google.bg/books?id=h1G2CgAAQBAJ&pg=PA327&redir_esc=y#v=onepage&q&f=false