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Question Number 160833 by Raxreedoroid last updated on 07/Dec/21

$$\underset{k=1}{\sum ^{\mathrm{\infty }}}\frac{\cos \left(\ln k\right)}{\sqrt{k}}$$$$\mathrm{divergespnt}\mathrm{or}\mathrm{convergent}?$$

Answered by mathmax by abdo last updated on 07/Dec/21

$$\mathrm{let}\mathrm{S}=\sum _{\mathrm{n}=1}^{\mathrm{\infty }}\frac{\cos \left(\mathrm{lnn}\right)}{\sqrt{\mathrm{n}}}\mathrm{the}\mathrm{sequence}{\mathrm{u}}_{\mathrm{n}}=\frac{1}{\sqrt{\mathrm{n}}}\mathrm{is}\mathrm{decreazing}\mathrm{to}$$$$\mathrm{we}\mathrm{can}\mathrm{prove}\mathrm{that}\mathrm{\exists }\mathrm{m}>0\mid \sum _{\mathrm{k}=1}^{\mathrm{n}}\cos \left(\mathrm{lnk}\right)\mid <\mathrm{m}$$$$\sum _{\mathrm{k}=1}^{\mathrm{n}}\cos \left(\mathrm{lnk}\right)⩽\sum _{\mathrm{k}=1}^{\mathrm{n}}\cos (\mathrm{k}-1)=\mathrm{Re}\left(\sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{e}}^{\mathrm{i}(\mathrm{k}-1)}\right)=....$$$$\mathrm{lnk}⩽\mathrm{k}-1?$$$$\phi \left(\mathrm{x}\right)=\mathrm{lnx}-\mathrm{x}+1(\mathrm{x}>1)\Rightarrow {\phi }^{{}^{\prime }}\left(\mathrm{x}\right)=\frac{1}{\mathrm{x}}-1=\frac{1-\mathrm{x}}{\mathrm{x}}<0$$$$\mathrm{x}1+\mathrm{\infty }$$$${\phi }^{{}^{\prime }}0-$$$$\phi 0\mathrm{decr}-\mathrm{\infty }\Rightarrow \mathrm{therem}\mathrm{dabel}\mathrm{dirichlet}\mathrm{the}\mathrm{serie}\mathrm{is}\mathrm{cv}..$$

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