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Question Number 82225 by M±th+et£s last updated on 19/Feb/20

Commented by mathmax by abdo last updated on 19/Feb/20

$$let{S}_n=\sum _{p=n+1}^{kn}\frac{1}{p}\Rightarrow S_n=\sum _{p=1}^{kn}\frac{1}{p}-\sum _{p=1}^n\frac{1}{p}=H_{kn}-H_n$$$$wehave{H}_{kn}=ln\left(kn\right)+\gamma +o\left(\frac{1}{n}\right)also{H}_n=ln\left(n\right)+\gamma +o\left(\frac{1}{n}\right)(n\to +\mathrm{\infty })$$$$H_{kn}-H_n=ln\left(k\right)+o\left(\frac{1}{n}\right)\Rightarrow {lim}_{n\to +\mathrm{\infty }}{S}_n=ln\left(k\right)$$

Commented by M±th+et£s last updated on 19/Feb/20

$$thankyousir$$

Commented by mathmax by abdo last updated on 19/Feb/20

$$youarewelcome.$$

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