A294970 Numerators of the partial sums for the Catalan constant A006752: Sum_{k=0..n} ((-1)^k)/(2*k+1)^2, n >= 0.
1, 8, 209, 10016, 91369, 10956424, 1863641881, 1854623872, 538015351033, 193637145687688, 194117166024913, 102476291858462752, 2566386635039604121, 23062916917686411464, 19421109407275720721849, 18642496069331249273291264
Offset: 0
Examples
The rationals r(n) begin: 1, 8/9, 209/225, 10016/11025, 91369/99225, 10956424/12006225, 1863641881/2029052025, 1854623872/2029052025, 538015351033/586396035225, 193637145687688/211688968716225, 194117166024913/211688968716225, 102476291858462752/111983464450883025, ... r(10^5) = 0.9159655942 (Maple 10 digits) to be compares with 0.91596559417... from A006752.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..575
- Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
- Eric Weisstein's World of Mathematics, Trigamma Function
Programs
-
Magma
[Numerator((&+[(-1)^k/(2*k+1)^2: k in [0..n]])): n in [0..20]]; // G. C. Greubel, Aug 22 2018
-
Mathematica
Table[Numerator[Sum[(-1)^k/(2*k+1)^2, {k,0,n}]], {n,0,20}] (* Vaclav Kotesovec, Nov 15 2017 *) -
PARI
for(n=0,20, print1(numerator(sum(k=0,n, (-1)^k/(2*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 22 2018
Formula
a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k/(2*k+1)^2 = (Zeta(2, 1/4) - Zeta(2,floor(n/2) + 5/4) - (Zeta(2, 3/4) - Zeta(2,floor((n-1)/2) + 7/4)))/16, Zeta(2, z) = Psi(1, z), with the Hurwitz Zeta function and the trigamma function Psi(1, z).
The limit n-> infinity of r(n) is the Catalan constant given in A006752; see in particular the formula (Zeta(2, 1/4) - Zeta(2, 3/4))/16.
Comments