Partial sums of $\sin(x)$

Asked May 05 '13 at 18:38

Active May 05 '13 at 18:56

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Is it true that $$ \left|\sum_{k=1}^n \sin(k) \right|\leq M $$ for every $n$?

I tried comparing this to the integral $\int_2^\infty \sin(x)dx$ but it is not monotone.

This is part of a problem that says if $A_n=a_1+a_2+\cdots+a_n$ and $|A_n|\leq M$ and $b_n$ is a decreasing sequence to zero, then $\sum a_n b_n$ converges. Then use this to show that $$ \sum_{n=2}^\infty \frac{\sin(n)}{\log(n)} $$

edited May 05 '13 at 18:56

MJD

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asked May 05 '13 at 18:38

Eager Student

Using the formula http://www.wolframalpha.com/input/?i=Sum+sin(kx)+for+k+%3D+1+to+n – Ma Ming May 05 '13 at 18:40
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Yes. The standard way to show this is to compute $\sum_{k=1}^n e^{ik}$ using the geometric series formula and then draw a conclusion on the real and imaginary parts. – Mark McClure May 05 '13 at 18:41
You could also argue as I do here. – David Mitra May 05 '13 at 18:45
Great! Thanks for that ! – Eager Student May 05 '13 at 18:50
This is know as Dirichlet's test. – Mhenni Benghorbal May 06 '13 at 05:49

Partial sums of $\sin(x)$

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