# Asking about reason of convergence of series on Apostol Mathematical analysis ( Chapter - Lebesgue Integral)

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

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.

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).
It is $$\sum_{n=1}^{\infty}n^{-s}\Gamma(s)=\Gamma(s)\sum_{n=1}^{\infty}\frac{1}{n^s}$$
• Worth emphasizing $s>1.$