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This is an after-chapter exercise of Dirichlet's Test.

Show that if the partial sum $S_n$ of the series $\displaystyle\sum_{k=1}^{\infty} a_k \leq Mn^r$, for some $r<1$, then the series $\displaystyle\sum_{k=1}^{\infty}\dfrac{a_k}{k}$ converges.

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  • $\begingroup$ I am thinking about to show the ∑_(k=1)^∞▒a_k /k^r',where r<r'<1,is bounded, then by Dirichlet's Test, the original one is convergent. But I just can't work it out. Asking for help. $\endgroup$ – Ray Xiang Apr 29 '13 at 5:37
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    $\begingroup$ Some of your formatting has gone astray. Maybe a cut and paste issue. Maybe you can fix it. Regards $\endgroup$ – Amzoti Apr 29 '13 at 5:48
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I suppose you are asking if the series $\sum_{k=1}^\infty\dfrac{a_k}{k}$ is convergent assuming that the partial sums of $\sum_{k=1}^\infty a_k$ are bounded by $Mn^r$. Is that right? If yes you have to assume also that $a_n\geq0$ (why?):

From $\sum_{k=1}^na_k\leq Mn^r$ it follows that $\sum_{k=1}^na_k\lt Mn$ (since $r\lt1$). Therefore $\sum_{k=1}^n\dfrac{a_k}{n}\lt M, \ \forall n\in\mathbb N$. Now use that $\frac1n\lt\frac1k$ for all $k\gt n$ to conclude that the series $\sum_{k=1}^\infty\dfrac{a_k}{k}$ is bounded...

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