The sum of the infinite series, 1/2^2 - 2/3^2 + 3/4^2 - 4/5^2 + ... is given as pi^2/12 - log 2 on pages 64-65 in the book "Summation of Series" by L. B. W. Jolley, 2nd ed., 1961, Dover Pubs. Inc. (the ^ symbol denotes exponentiation in the above series and sum).
For most of the series in his book, he lists a source (reference), but does not do so for this one.
My question is: Who proved this originally and where and when was it first published? I would like to see a proof for this, especially the original one (in English), but I will be satisfied with any proof you can provide. However, I would also like to know who first proved it and when, along with a reference to where it was published.
Please see the attachment.
Actually, the infinite series you want to discuss in this posting is not a special sequence. It is a combination of two famous sequences and those two famous ...
This explains the result of a given sum of an infinite series. The proof of a sum for given infinite series of constants are determined. A closed form analysis is provided.