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Show that if $x \neq 0,\pm 2 \pi,\pm 4 \pi, \dots$, then

$$\frac{1}{\tan(x/2)}=2 \sum_{j=1}^{\infty}\sin(jx)$$

in Cesàro way/sense. Some hint whether to manipulate

$\sum_{j=1}^{\infty}a_j(x)=\sum_{j=1}^{\infty}\sin(jx) \tag1$

into (using partial sum of ($1$)) $\frac{1}{n}\sum_{j=1}^{n}\sin(jx)= \dots$

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Would someone edit it better if I have lost some thing when I edited it lately. I don't know why it does not put $\sum_{j=1}^{\infty}a_j(x)$ after $\sum_{j=1}^{\infty}sin(jx)$ –  laovultai Oct 14 '12 at 21:21

1 Answer 1

up vote 2 down vote accepted

HINT

$$S_n(x) = \sum_{k=1}^n \sin(kx) = \dfrac{\cos(x/2) - \cos((n+2)x/2)}{2\sin(x/2)}$$ To prove this multiply, $S_n(x)$ by $\sin(x/2)$ and make use of the fact that $$\sin(A) \sin(B) = \dfrac{\cos(A-B) - \cos(A+B)}2$$ and do telescopic cancellation.

EDIT $$\dfrac{\displaystyle \sum_{n=1}^{N} S_n(x)}N = \dfrac1{2 \tan(x/2)} - \dfrac{\displaystyle \sum_{n=1}^{N} \cos((n+2)x/2)}{2N \sin(x/2)}$$

The second term without the $N$ in the denominator is bounded (Why? One way is to evaluate the sum in a similar spirt as above or write it as exponential and use geometric series to see that it is bounded when $x \neq k \pi$). Hence, if you take the limit as $N \to \infty$, the second term will tend to $0$.

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But how would you use Cesàro method i.e. $\frac{1}{n} \sum_{j=1}^{n} sin(jx) \rightarrow a$ as $j \rightarrow \infty$, where $a \in N$ with that proof? What is telescopic cancellation? I know that it is A-B+(B-C)+(C-D)=A-D, but hard time in figuring out how to implement it in this case? –  laovultai Oct 14 '12 at 20:12
    
@alvoutila What you want to show is the following. If $$s_n = \sum_{k=1}^{n} a_k$$ then you want to look at $$S_N = \dfrac{\displaystyle \sum_{n=1}^{N} s_n}N$$ and show that the limit of $S_N$ exists. (This is the Cesaro way.) I have added details to the post above and hope now it is clear. –  user17762 Oct 14 '12 at 20:32
    
If it suites you I come back to comment because I have to first set problem which I try to solve and then compare your ideas with what I have achieved. And maybe then I can formulate some questions for you to answer. I understood your comment above, because I know what is Cesàro way as presented in lectures. But Hint and Edit part are not yet fully understood. –  laovultai Oct 14 '12 at 21:25
    
firstly I think that you don't have to care constant 2 at all. I mean it is all about $\sum_{n=1}^{N}s_n$, where $s_n=sin(nx)$? –  laovultai Oct 14 '12 at 21:43

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