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While studying Apostol Mathematical analysis I am unable to find reason of following argument whose picture follows : enter image description here

In first line of last paragraph can someone please tell how author wrote " the series on right being convergent " ? Can someone please tell how this particular series is convergent?

Any help will be really appreciated.

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2 Answers 2

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From the fourth equation of your image $$n^{-s}\Gamma(s) = \int_0^{+\infty}e^{-nx}x^{s-1}ds$$ We want to know the nature of the series $\sum_{n\geq 1}\int_0^{+\infty}e^{-nx}x^{s-1}ds$ or, equivalently, of the series $\sum_{n\geq 1}n^{-s}\Gamma(s)$. Since $\Gamma(s)\in\mathbb{R}$ is constant, this series converges if, and only if $\sum_{n\geq 1}n^{-s}$ converges. It is a classical result in analysis that this last series converges if, and only if, $s>1$ (this is reminded at the beginning of the penultimate paragraph of your image).

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It is $$\sum_{n=1}^{\infty}n^{-s}\Gamma(s)=\Gamma(s)\sum_{n=1}^{\infty}\frac{1}{n^s}$$

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  • $\begingroup$ Worth emphasizing $s>1.$ $\endgroup$
    – zhw.
    Jun 23, 2020 at 15:24
  • $\begingroup$ Yes, it is in line above. $\endgroup$
    – zkutch
    Jun 23, 2020 at 15:29

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