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Another proof of Euler’s formula for \(\zeta(2k)\). (English) Zbl 1223.40001

For Euler’s famous evaluation in question, there is an abundance of proofs. The authors give another one which they “believe” to be “simpler than those cited” in their long list of references. Here, Euler’s formula results from comparing two series representations of \(\int\tan xy\text{ d}x\), \(|x|\leq \pi/2\), \(|y|<1\), by unspecified reference to “Fubini’s theorem”.
Reviewer’s remark: At the first representation, the index \(k\) instead of \(n\) would be appropriate for later identification. – In the last equation line, the factor has to change sides.

MSC:

40A25 Approximation to limiting values (summation of series, etc.)
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References:

[1] Tom M. Apostol, Another elementary proof of Euler’s formula for \?(2\?), Amer. Math. Monthly 80 (1973), 425 – 431. · Zbl 0267.10050
[2] Raymond Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067 – 1086. · Zbl 0293.10001
[3] Árpád Bényi, Finding the sums of harmonic series of even order, College Math. J. 36 (2005), no. 1, 44 – 48. · Zbl 02365445
[4] Bruce C. Berndt, Elementary evaluation of \?(2\?), Math. Mag. 48 (1975), 148 – 154. · Zbl 0303.10038
[5] L. Carlitz, A recurrence formula for \?(2\?), Proc. Amer. Math. Soc. 12 (1961), 991 – 992.
[6] Chapman, R., Evaluating \( \zeta (2)\), preprint, http://www.maths.ex.ac.uk /\( \sim \)rjc/etc/zeta2.pdf
[7] Chen, X., Recursive formulas for \( \zeta (2k)\) and \( L(2k-1)\), College Math. J. 26 (1995) 372-376.
[8] Ming Po Chen, An elementary evaluation of \?(2\?), Chinese J. Math. 3 (1975), no. 1, 11 – 15. · Zbl 0361.10015
[9] Ji Chungang and Chen Yonggao, Euler’s Formula for ?(2k), Proved by Induction on k, Math. Mag. 73 (2000), no. 2, 154 – 155.
[10] T. Estermann, Elementary evaluation of \?(2\?), J. London Math. Soc. 22 (1947), 10 – 13. · Zbl 0029.39403
[11] Rolf M. Hovstad, The series \sum _{\?=1}^{\infty }1/\?^{2\?}, the area of the unit circle and Leibniz’ formula, Nordisk Mat. Tidskr. 20 (1972), 92 – 98, 120. · Zbl 0253.40003
[12] Morris Kline, Euler and infinite series, Math. Mag. 56 (1983), no. 5, 307 – 314. · Zbl 0526.01015
[13] Huan-Ting Kuo, A recurrence formula for \?(2\?), Bull. Amer. Math. Soc. 55 (1949), 573 – 574. · Zbl 0032.34501
[14] Thomas J. Osler, Finding \?(2\?) from a product of sines, Amer. Math. Monthly 111 (2004), no. 1, 52 – 54. · Zbl 1122.11309
[15] Neville Robbins, Revisiting an Old Favorite: ?(2m), Math. Mag. 72 (1999), no. 4, 317 – 319.
[16] E. L. Stark, \sum ^{\infty }_{\?=1}\?^{-\?}, \?=2,3,4\cdots, once more, Math. Mag. 47 (1974), 197 – 202. · Zbl 0291.40004
[17] Hirofumi Tsumura, An elementary proof of Euler’s formula for \?(2\?), Amer. Math. Monthly 111 (2004), no. 5, 430 – 431. · Zbl 1080.11061
[18] G. T. Williams, A new method of evaluating \?(2\?), Amer. Math. Monthly 60 (1953), 19 – 25. · Zbl 0050.06803
[19] Kenneth S. Williams, On \sum _{\?=1}^{\infty }(1/\?^{2\?}), Math. Mag. 44 (1971), 273 – 276. · Zbl 0224.40008
[20] Zhang Nan Yue and Kenneth S. Williams, Application of the Hurwitz zeta function to the evaluation of certain integrals, Canad. Math. Bull. 36 (1993), no. 3, 373 – 384. · Zbl 0791.11045
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