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

Question

$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

16

Report

Answer

The answer is accepted.

Join Matchmaticians Affiliate Marketing Program to earn up to 50% commission on every question your affiliated users ask or answer.

Answer 1

Answers can be viewed only if

The questioner was satisfied and accepted the answer, or
The answer was disputed, but the judge evaluated it as 100% correct.

View the answer

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

4.4K

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

Answer

Related Questions

Search