
06164064
j
2013c.00433
Wright, Cory
Osler, Thomas J.
Four derivations of an interesting bilateral series generalizing the series for zeta of 2.
Int. J. Math. Educ. Sci. Technol. 44, No. 3, 456461 (2013).
2013
Taylor \& Francis, Abingdon, Oxfordshire
EN
F65
I45
I55
bilateral series
fractional derivatives
zeta function
doi:10.1080/0020739X.2012.714495
Summary: We present four derivations of the closed form of the partial fractions expansion $$\pi \left(\dfrac{\cot \pi a}{ba}\dfrac{\cot \pi b}{ab}\right) = \sum_{n=\infty}^{\infty}\dfrac{1}{(n+a)(n+b)}.$$ This interesting series is a generalization of the series $\frac{\pi ^2}{6}= \sum_{n=1}^{\infty} \frac{1}{n^2} = \zeta(2)$ made famous by Euler.