Friday, 18 October 2013

calculus - How prove this suminftyn=1fraczeta2n4=zeta2(3)frac13zeta(6)

show that
n=1ζ2n4=ζ2(3)13ζ(6)



where
ζm=nk=11km,ζ(m)=k=11km


is true?
because This result is my frend tell me.




This problem have someone research it?Thank you



my some idea:
ζ3(3)=(n=01(n+1)3)2=n=0nk=01(k+1)3(nk+1)3



and use
1(k+1)(nk+1)=1n+2(1k+1+1nk+1)


and (a+b)3=a3+3a2b+3ab2+b3

and
n=1Hn(n+1)5=12(5ζ(6)2ζ(2)ζ(4)ζ2(3))


But is very ugly, someone have other nice methods? Thank you .

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...