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Let $A_n=\sum\limits_{k=1}^n \sin k $ , show that there exists $M>0$ , $|A_n|<M $ for every $n$ .

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What are your thoughts on the problem? What have you tried? Just so you know, it is also considered a bit rude to post in the imperative. It is quite alright to just ask a question if you have one instead of giving orders. – JavaMan Aug 25 '11 at 2:09
up vote 36 down vote accepted

Note that $$2(\sin k)(\sin(0.5))=\cos(k-0.5)-\cos(k+0.5).$$ This is obtained by using the ordinary expression for the cosine of a sum.

Add up, $k=1$ to $n$. On the right, there is mass cancellation. We get $$\cos(0.5)-\cos(n+0.5).$$ Thus our sum of sines is $$\frac{\cos(0.5)-\cos(n+0.5)}{2\sin(0.5)}.$$ We can now obtain the desired bound for $|A_n|$. For example, $2$ works, but not by much.

We could modify the appearance of the above formula by using the fact that $\cos(0.5)-\cos(n+0.5)=2\sin(n/2)\sin(n/2+0.5)$.

Generalization: The same idea can be used to find a closed form for $$\sum_{k=0}^{n-1} \sin(\alpha +k\delta).$$ Sums of cosines can be handled in a similar way.

Comment: This answer was written up because the OP, in a comment, asked for a solution that only uses real functions. However, summing complex exponentials, as in the solution by @Eric Naslund, is the right way to handle the problem.

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Hi @andrenicolas, how does Eric Naslund's answer below show the boundedness of the partial sums of sin(x)? Looking at the geometric sum formula, the exponentials will have modulus = 1, and the sum formula gives a (1-1) on the denominator, when going for an upper bound, and the function blows up. Where is my mistake? Thanks so much, – User001 Nov 25 '15 at 2:31
Take one of the geometric series, say $\sum_1^K e^{in}$. This is $\frac{1-e^{i(K+1)}}{1-e^i}$. The top has norm $\le 2$, and the bottom has fixed non-zero norm, so we have boundedness. – André Nicolas Nov 25 '15 at 2:40
Hi @andrenicolas, if the sum starts at n=1, the sum formula should then give $\large \frac{e^i (1-e^{i(K+1)})}{(1-e^i)}$ right? We pick up a scalar $\large e^i$, I think, but I just want to confirm -- I see so many different versions of the geometric sum formula that I can get a little confused sometimes -- thanks so much, – User001 Nov 25 '15 at 2:51
Nevermind, I am working it out following the derivations from Wikipedia -- thanks @andrenicolas :-) – User001 Nov 25 '15 at 3:11
Oops, I did not notice it started at $1$, the sum is then $\frac{e^i(1-e^{iK})}{1-e^i}$, like yours, marginally different. – André Nicolas Nov 25 '15 at 3:16

Hint: Since $$\sin x = \frac{e^{ix}-e^{-ix}}{2i}$$ we can rewrite $$\sum_{n=1}^{K} \sin n = \frac {1}{2i}\sum_{n=1}^K (e^i)^n-\frac {1}{2i}\sum_{n=1}^K (e^{-i})^n$$ and both of these are geometric series.

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That's good. Can you show me a method just with real function. – Leitingok Aug 25 '11 at 2:26
But $|| e^i|| = 1$, so it is not a geometric series! (I mean, it is but it does not converge).. Am I wrong? – Ant Apr 8 '14 at 18:05
@Ant: It's a finite series so it does converge. It is true that the infinite series $\sum_{n=1}^\infty e^{inx}$ does converge, but that is not what appears here. – Eric Naslund Apr 8 '14 at 18:10
of course.. but how does this solve the problem? I mean if we're talking about finite series every series converge. $\sum_0^k \sin(n)$ is a finite series hence converge. So why rewriting the partial sum of $\sin(n)$ that way? How does it help exactly? Thank you for your reply! :-) – Ant Apr 8 '14 at 18:14
@Ant: The question asked to show that $\sum_{n=0}^k \sin(n)$ is bounded for all $k$. Boundedness is a less stringent condition than convergence. (All convergent sequences are bounded but not vice versa) – Eric Naslund Apr 8 '14 at 18:36

A more general formula would be:

$$A_n=\sum_{k=1}^{n} \sin k\theta = \frac{\sin\theta+\sin n\theta-\sin(n+1)\theta}{2(1-\cos\theta)}$$ So $A_n$ is clearly bounded (just simply check the case where $\theta=1$).

The formula can be proved by induction using the trig identity: $\sin\alpha+\sin\beta=2\sin(\frac{\alpha+\beta}{2})\cos(\frac{\alpha-\beta}{2})$.

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