Wednesday, January 27, 2016

sequences and series - limit limlimitsntoinftyleft(sumlimitsni=1frac1sqrti2sqrtnright)



Calculate below limit limn(ni=11i2n)


Answer



As a consequence of Euler's Summation Formula, for s>0, s1 we have nj=11js=n1s1s+ζ(s)+O(|ns|),

where ζ is the Riemann zeta function. In your situation, s=1/2, so nj=11j=2n+ζ(1/2)+O(n1/2),
and we have the limit limn(nj=11j2n)=limn(ζ(1/2)+O(n1/2))=ζ(1/2).


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