Wednesday, 14 May 2014

calculus - Interesting Harmonic Sum sumkgeq1frac(1)k1k2H(2)k



Here http://integralsandseries.prophpbb.com/topic119.html




We came across the following harmonic sum



k1(1)k1k2H(2)k



Note that we define



H(2)k=kn11n2



Also we have




ψ1(k+1)=ζ(2)H(2)k



Any ideas how to evaluate (1) ?


Answer



A related problem. You can have the following identity




k=1(1)k1H(2)kk2=3716ζ(4)+2k=1(1)kHkk30.7843781621.




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