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Question Number 86113 by Tony Lin last updated on 27/Mar/20

$$\left(1\right)Determinethefollowing$$$$ifitisconvergentordivergent$$$$\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{sin\left(n\right)}{n}$$$$\left(2\right)\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{sin\left(n^p\right)}{n^p},pϵ\mathbb{R},findtherange$$$$ofpwhenitisconvergent$$

Commented by Serlea last updated on 27/Mar/20

$$$$$$$$$$\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{sin\left(n\right)}{n}$$$$\mathrm{let}{\mathrm{U}}_{\mathrm{n}}=\frac{\sin \left(\mathrm{n}\right)}{\left(\mathrm{n}\right)}$$$$=\left(\frac{1}{\mathrm{n}}\right)\sin \left(\mathrm{n}\right)$$$${\mathrm{V}}_{\mathrm{n}}=\frac{\mathrm{n}}{\mathrm{n}}=1$$$${\lim }_{\mathrm{n}\to \mathrm{\infty }}\frac{{\mathrm{U}}_{\mathrm{n}}}{{\mathrm{V}}_{\mathrm{n}}}={\lim }_{\mathrm{n}\to \mathrm{\infty }}\frac{\sin \left(\mathrm{n}\right)}{\mathrm{n}}=1$$$$\mathrm{\Sigma }{\mathrm{U}}_{\mathrm{n}}\mathrm{and}\mathrm{\Sigma }{\mathrm{V}}_{\mathrm{n}}\mathrm{but}\mathrm{converges}\mathrm{and}\mathrm{diverges}$$$${\mathrm{It}}^{\prime }\mathrm{s}\mathrm{diverent}\mathrm{from}\mathrm{the}\mathrm{P}-\mathrm{series}\mathrm{test}$$$$(\mathrm{p}=0<1)$$$$$$$$2)\mathrm{\Sigma }\frac{\sin \left({\mathrm{n}}^{\mathrm{p}}\right)}{{\mathrm{n}}^{\mathrm{p}}}$$$${\mathrm{U}}_{\mathrm{n}}=\frac{\sin \left({\mathrm{n}}^{\mathrm{p}}\right)}{{\mathrm{n}}^{\mathrm{p}}}$$$${\mathrm{V}}_{\mathrm{n}}=\frac{{\mathrm{n}}^{\mathrm{P}}}{{\mathrm{n}}^{\mathrm{p}}}=1$$$${\mathrm{Lim}}_{\mathrm{n}\to \mathrm{\infty }}\frac{{\mathrm{U}}_{\mathrm{n}}}{{\mathrm{V}}_{\mathrm{n}}}={\lim }_{\mathrm{n}\to \mathrm{\infty }}\frac{\sin \left({\mathrm{n}}^{\mathrm{p}}\right)}{{\mathrm{n}}^{\mathrm{p}}}\mathrm{let}\mathrm{t}={\mathrm{n}}^{\mathrm{p}}$$$$=1$$$$$$$$\mathrm{P}=1$$$$\mathrm{Convergent}\mathrm{when}\mathrm{p}⩾1$$

Commented by mathmax by abdo last updated on 27/Mar/20

$$1)\sum _{n=1}^{\mathrm{\infty }}\frac{sin\left(n\right)}{n}convergesduetoabeldirichlettheorem$$$$u_n=\frac{1}{n}isdeceasingto0and\mathrm{\exists }m>0/\mid \sum _{k=0}^{n}sink\mid <m$$

Commented by mathmax by abdo last updated on 27/Mar/20

$$\mid \sum _{n=1}^{\mathrm{\infty }}\frac{sin\left(n^p\right)}{n^p}\mid ⩽\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^p}soifp>1thisserieconverges...$$

Commented by Serlea last updated on 27/Mar/20

$$\mathrm{I}\mathrm{think}\mathrm{you}\mathrm{are}\mathrm{wrong}\mathrm{bro}$$

Commented by mathmax by abdo last updated on 27/Mar/20

$$iamnotwrongrevisecoursesofseries....$$

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