For any positive integer n [1], the sum in odd form related to the Riemann zeta function is expressed as follows:
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The first 15 values in closed-form of the sum involving the reciprocals of odd integers raised to even powers are:
1. 1 + 1 ⁄ 32 + 1 ⁄ 52 + 1 ⁄ 72 + ... = π2 ⁄ 8.
2. 1 + 1 ⁄ 34 + 1 ⁄ 54 + 1 ⁄ 74 + ... = π4 ⁄ 96.
3. 1 + 1 ⁄ 36 + 1 ⁄ 56 + 1 ⁄ 76 + ... = π6 ⁄ 960.
4. 1 + 1 ⁄ 38 + 1 ⁄ 58 + 1 ⁄ 78 + ... = 17π8 ⁄ 161280.
5. 1 + 1 ⁄ 310 + 1 ⁄ 510 + 1 ⁄ 710 + ... = 31 π10 ⁄ 2903040.
6. 1 + 1 ⁄ 312 + 1 ⁄ 512 + 1 ⁄ 712 + ... = 691 π12 ⁄ 638668800.
7. 1 + 1 ⁄ 314 + 1 ⁄ 514 + 1 ⁄ 714 + ... = 5461 π14 ⁄ 49816166400.
8. 1 + 1 ⁄ 316 + 1 ⁄ 516 + 1 ⁄ 716 + ... = 929569 π16 ⁄ 83691159552000.
9. 1 + 1 ⁄ 318 + 1 ⁄ 518 + 1 ⁄ 718 + ... = 3202291 π18 ⁄ 2845499424768000.
10. 1 + 1 ⁄ 320 + 1 ⁄ 520 + 1 ⁄ 720 + ... = 221930581 π20 ⁄ 1946321606541312000.
11. 1 + 1 ⁄ 322 + 1 ⁄ 522 + 1 ⁄ 722 + ... = 4722116521 π22 ⁄ 408727537373675520000.
12. 1 + 1 ⁄ 324 + 1 ⁄ 524 + 1 ⁄ 724 + ... = 56963745931 π24 ⁄ 48662619743783485440000.
13. 1 + 1 ⁄ 326 + 1 ⁄ 526 + 1 ⁄ 726 + ... = 14717667114151 π26 ⁄ 124089680346647887872000000.
14. 1 + 1 ⁄ 328 + 1 ⁄ 528 + 1 ⁄ 728 + ... = 2093660879252671 π28 ⁄ 174221911206693634572288000000.
15. 1 + 1 ⁄ 330 + 1 ⁄ 530 + 1 ⁄ 730 + ... = 86125672563201181 π30 ⁄ 70734095949917615636348928000000.
(12/29/2009)
The Values of Riemann Zeta Function for Positive Even Integers
References
[1] Sury, B. Bernoulli numbers and the riemann zeta function. Reson 8, 54–62 (2003). https://doi.org/10.1007/BF02834403.
Hyperlink: http://seriesmathstudy.com/sms/evenzetainoddform.
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