I ask this because I noticed the partial sum ∑xn=11√n is very close to 2√x, 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(n−1), an application of telescoping series to partial sums.
Thus,
limx→∞(2√x−x∑n=11√n)=limx→∞(2√0+x∑n=1(2√n−2√n−1)−x∑n=11√n)=∞∑n=1(2√n−2√n−1−1√n)
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 x∑n=11√n=H(12)x
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