Does $\sum_{n=1}^\infty \sin \frac a n$ converge?

Asked May 20 '23 at 07:58

Active May 20 '23 at 08:00

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Examine for convergence the series $\sum_{n=1}^\infty \sin \frac a n$.

The series is divergent if we consider $a_n = \sin \frac a n$, $b_n =\frac{ a}{n}$ then apply the comparison test.

But this series is not a series with positive terms.

According to the question, it is an alternating series.

How can we deal with it as an alternating series?

edited May 20 '23 at 08:00

Kenny Wong

asked May 20 '23 at 07:58

Riya Garg

Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers. – Another User May 20 '23 at 07:59
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It is not alternating. – David Mitra May 20 '23 at 08:02
Why is it not alternating? – Riya Garg May 20 '23 at 08:12
Alternating means every positive term is always followed by a negative term and vice versa. – balddraz May 20 '23 at 08:29
So which type of series is this and how to test it's convergence ? – Riya Garg May 20 '23 at 08:37
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If $a>0$, your series $S$ has only positive terms; convergence can be concluded from what you wrote. If $a<0$, look at $-S$. – David Mitra May 20 '23 at 08:55
@DavidMitra: (... eventually ...) – metamorphy May 20 '23 at 09:31
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This answers (a generalization of) your question: Uniform convergence of $\sum n^\alpha \sin{\frac{x}{n}}$ – Anne Bauval May 20 '23 at 09:34
@metamorphy oopsie! – David Mitra May 20 '23 at 11:08

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