Sunday, 11 June 2017

sequences and series - What is limxtoinfty2sqrtxsumxn=11oversqrtn?





I ask this because I noticed the partial sum xn=11n is very close to 2x, so close in fact that it appears their difference approaches a constant value, like Hx and lnx. However, when I put this limit as is into Wolfram, it said the limit diverged.



But, I found a way to transform the limit into an infinite sum, by using the transformation f(x)=f(0)+xn=1f(n)f(n1), an application of telescoping series to partial sums.



Thus,
limx(2xxn=11n)=limx(20+xn=1(2n2n1)xn=11n)=n=1(2n2n11n)



This sum converges according to Wolfram by comparison test, and according to me by the integral test, but what does it converge to? It converges incredibly slowly; my best guess is 1.458


Answer



May be, we could use generalized harmonic numbers since xn=11n=H(12)x

and use the asymptotics for large values of n H(12)x=2x+ζ(12)+12x+O(1x3/2)


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