Let $s \in \Bbb C$ be a complex number with $\operatorname*{Re}(s)>0$ and the series $\sum_{n=1}^\infty \left| \frac{1}{(n+1)^s}-\frac{1}{n^s} \right|$
How can i prove that it is convergent?
Thanks
How to prove the Convergence of this Series
2
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sequences-and-series
complex-analysis
1 Answers
1
We have
$$\frac1{(n+1)^s}=\frac1{n^s}\left(1+\frac1n\right)^{-s}=\frac1{n^s}\left(1-\frac{s}{n}+o(n^{-1})\right)$$ so we see that the given series has the same nature as the series $\sum\limits_n \left\vert\frac{1}{n^{s+1}}\right\vert=\sum_\limits n\frac{1}{n^{\operatorname{Re}(s)+1}}$ which is a convergent Riemann series ($\operatorname{Re}(s)+1>1)$.