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.
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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 ...
Solution Summary
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.
... a) The summation becomes E(X) = sum c^k * p(x) = sum (c/2)^k = c/2 + (c/2)^2 + (c/2)^3 + ... This is an infinite geometric series with common ratio c/2. Read ...
... 2 rad can be represented by the infinite series S = sum... 2n+1)!] Let its partial sum be SN = sum (0 to ... Use the Ratio test to show the series converges absolutely ...
Define the set R[[X]] of formal power series in the indeterminate X with coefficients from R to be all formal infinite sums sum(a_nX^n)=a_0 +a_1X+a_2X^2 ...
... the solution obtained via separation of variables is an infinite series. ... solution to the heat equation on the infinite interval. ... but v(x.0) = sum bn sin (nPI x ...
... the function is decomposed into eigenfunctions series, how to ... The normalized orthogonal eigenfunctions of the infinite well are ... m we can bring it into the sum: ...
... Therefore: (2.3) The domain is There is infinite number of ... can write in terms of its Laurent series: (5.2) Since ... For the sum converges to zero for any finite ...
... the device is left on for an infinite amount of ... would be the power dissipated by the series circuit described ... that the total current consists of the sum of the ...
... a infinite geometric series from second term onwards. Its first term is 5/100 and common ratio is 1/10 5 5 102 = 105 + 5 = 945 + 5 = 950 = 9 2 Hence the sum is ...
... has cylindrical symmetry and can be considered infinite in the z ... we can bring the integrals into the sum: (1.31) Due to ... is the term in the cosine series where n ...
