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Question Number 30758 by abdo imad last updated on 25/Feb/18

$$find{lim}_{n\to \mathrm{\infty }}(\frac{1}{n+1}+\frac{1}{n+2}+....+\frac{1}{n+p})pfixed{fromN}^{★}$$$$$$

Commented by abdo imad last updated on 28/Feb/18

$$letput{u}_n=\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{n+p}$$$$u_n=H_{n+p}-H_n=ln(n+p)+\gamma +o\left(\frac{1}{n}\right)-ln\left(n\right)-\gamma -o^{{}^{\prime }}\left(\frac{1}{n}\right)$$$$=ln\left(\frac{n+p}{n}\right)+o{\left(\frac{1}{n}\right)}_{n\to \mathrm{\infty }}\to 0\Rightarrow {lim}_{n\to \mathrm{\infty }}{u}_n=0$$

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