Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

Question Number 138068 by Ñï= last updated on 09/Apr/21

                  Σ_(n=1) ^∞ x^(2n) ζ(2n)=(1/2)−((πx)/2)cot πx

n=1x2nζ(2n)=12πx2cotπx

Answered by Dwaipayan Shikari last updated on 09/Apr/21

Σ_(n=1) ^∞ Σ_(k=1) ^∞ (x^(2n) /k^(2n) )=Σ_(k=1) ^∞ (x^2 /(k^2 −x^2 ))=(x/2)Σ_(k=1) ^∞ (1/(k−x))−(1/(k+x))  =(x/2)(ψ(x+1)−ψ(1−x))=(x/2)(ψ(x)+(1/x)−ψ(1−x))  =(1/2)−((xπ)/2)cot(πx)               ψ(x)−ψ(1−x)=−πcot(πx)

n=1k=1x2nk2n=k=1x2k2x2=x2k=11kx1k+x=x2(ψ(x+1)ψ(1x))=x2(ψ(x)+1xψ(1x))=12xπ2cot(πx)ψ(x)ψ(1x)=πcot(πx)

Terms of Service

Privacy Policy

Contact: info@tinkutara.com