$Show that $\sum_{n=1}^{\infty} \frac{\sin n}{n}$ is convergent$

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$Show that $\sum_{n=1}^{\infty} \frac{\sin n}{n}$ is convergent$

Show that $\sum_{n=1}^{\infty} \frac{\sin n}{n}$ is convergent.

I was trying to use the Dirichlet test, but couldn't verify that the conditions of the Dirichlet test are satisfied.

Calculus Sequences and Series Trigonometric Functions

Erica89

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We have
\[|\sin (1) \sum_{n=1}^{k}\sin n|=|\sum_{n=1}^{k}\sin (1) \sin n|\]
\[=\frac{1}{2}|\sum_{n=1}^{k}(\cos (n-1)-\cos(n+1))|=\frac{1}{2}|(\cos (0)+\cos (1)-\cos(k)-\cos(k+1))|\]
\[\leq \frac{1}{2} \times 4=2.\]
Hence
\[| \sum_{n=1}^{k}\sin n| \leq \frac{2}{\sin (1)}.\]
Also
\[\lim_{n\rightarrow} \frac{1}{n}=0.\]
Therefore by Dirichlet test
\[\sum_{n=1}^{\infty}\frac{\sin n}{n}\]
is convergent.

Erdos

4.8K

$Show that $\sum_{n=1}^{\infty} \frac{\sin n}{n}$ is convergent$

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