How to calculate convergence of a series related to the Euler's constant?

Asked Oct 28 '23 at 02:40

Active Oct 28 '23 at 02:40

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I understand that

$$ \sum_{r=1}^n \frac{1}{r} \approx \int_{1}^n \frac{dx}{x} = \log n $$

I would like to verify whether the following holds true as well, for a series that is related to the above series:

$$ \sum_{r=1}^n \frac{1}{r^{1/2}} \approx (log n)^{1/2} $$

I am doubtful whether this will work though. I tried various ways to verify this but I am probably not using the correct fundamental properties required here. Any guidance on how to calculate $$ \sum_{r=1}^n \frac{1}{r^{1/2}} $$ will be very helpful. Thank you so much.

asked Oct 28 '23 at 02:40

Shatarupa18

1

You might double check whether this has already been asked on MSE; for example, it looks to me like MSE 642617 is related. – Benjamin Dickman Oct 28 '23 at 02:45
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your approximation must be $\sum_{k=1}^n \frac1{k^{1/2}}\approx \int_{1}^n \frac1{x^{1/2}},d x=2\sqrt{n}-2$ instead of what you wrote – Masacroso Oct 28 '23 at 03:04
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the sum is approximately $2 \sqrt n + \zeta(1/2) + \frac{1}{2 \sqrt n}$ where $\zeta(1/2) = -1.4603545088095868128894991525152980125...$ – Will Jagy Oct 28 '23 at 03:50
Thank you! I will work through this... – Shatarupa18 Oct 30 '23 at 04:55

How to calculate convergence of a series related to the Euler's constant?

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