Monday, 21 July 2014

real analysis - Advanced Sum: Compute sumin=1nftyfracH2nH(2)n(2n+1)2

How to prove





n=1H2nH(2)n(2n+1)2=43ln32ζ(2)72ln22ζ(3)2116ζ(2)ζ(3)+71364ζ(5)415ln528ln2Li4(12)8Li5(12)




where H(q)n=nk=11nq is the harmonic number, Lir(x)=n=1xnnr is the polylogarithm function and ζ is the Riemann zeta function.



This problem is proposed by Cornel with no solution submitted.







My trial



By applying integration by parts we have



10x2n(Li2(x)ζ(2)) dx=H2n(2n+1)21(2n+1)3



now multiply both sides by H(2)n then sum both sides from n=1 to we get




10(Li2(x)ζ(2))n=1H(2)nx2n dx=n=1H2nH(2)n(2n+1)2n=1H(2)n(2n+1)3



10(Li2(x)ζ(2))Li2(x2)1x2 dx=n=1H2nH(2)n(2n+1)2n=1H(2)n(2n+1)3



I managed to find the blue sum using Abel's summation. As for the integral, I tried integration by parts but still resistant.




QUESTION



Any idea how to crack the integral or a different approach to find the target sum?





Thanks.

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...