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I am to find the sum of a series that takes this format

$ \sum_{i=1}^{n}\frac{1}{i^\beta} $

$ \beta $ is a positive real number

How to approach the partial sum of the above series and can obtain its estimation as below

$ \approx\frac{n^{1-\beta}}{1-\beta} $

Thanks a lot for any help

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1 Answer 1

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Hint:compare with the integral $$\int_1^n \frac{dx}{x^\beta}.$$

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  • $\begingroup$ your post reminds me of this one: math.stackexchange.com/a/208830 $\endgroup$
    – 311411
    Commented Feb 9, 2022 at 17:18
  • $\begingroup$ @311411 It’s really an old question $\endgroup$
    – mio
    Commented Feb 9, 2022 at 17:26

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