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## Main Question or Discussion Point

I really need help with this exercise (it's from a course in basic fourier analysis). It consists of two parts:

(i) Let [Tex] s_0 = 1/2 [/Tex] and [Tex] s_n = 1/2 + \sum_{j=1}^{n}\cos(jx) [/Tex] for [Tex] n \geq 1 [/Tex]. By writing [Tex] s_n = \left(\sum_{j=-n}^{n}e^{ijx}\right)/2 [/Tex] and summing geometric series show that [Tex] (n+1)^{-1}\sum_{j=0}^{n}s_j \rightarrow 0 [/Tex] as [Tex] n \rightarrow \infty [/Tex] for all [Tex] x \neq 0~mod~2\pi [/Tex], and so

[Tex] 0 = 1/2 + \sum_{j=1}^{\infty}\cos(jx) [/Tex] in the Cesáro sense.

(ii) Show similarly that, if [Tex] x \neq 0~mod~2\pi [/Tex], then

[Tex] cot(x/2) = 2\sum_{j=1}^{\infty}\sin(jx) [/Tex] in the Cesáro sense.

In (i) I have tried to write out two geometric series and summing them, but I can't get the desired result. I have no idea on (ii).

"in the Cesáro sense" means (i think) that the average of a given sequence [Tex] s_0,s_1,s_2,\ldots [/Tex] converges against a given limit L (the sequence itself doesn't nescessarily) - that is, the sequence [Tex] s_0, (s_0 + s_1)/2, (s_0 + s_1 + s_2)/3,\ldots \rightarrow L [/Tex].

(i) Let [Tex] s_0 = 1/2 [/Tex] and [Tex] s_n = 1/2 + \sum_{j=1}^{n}\cos(jx) [/Tex] for [Tex] n \geq 1 [/Tex]. By writing [Tex] s_n = \left(\sum_{j=-n}^{n}e^{ijx}\right)/2 [/Tex] and summing geometric series show that [Tex] (n+1)^{-1}\sum_{j=0}^{n}s_j \rightarrow 0 [/Tex] as [Tex] n \rightarrow \infty [/Tex] for all [Tex] x \neq 0~mod~2\pi [/Tex], and so

[Tex] 0 = 1/2 + \sum_{j=1}^{\infty}\cos(jx) [/Tex] in the Cesáro sense.

(ii) Show similarly that, if [Tex] x \neq 0~mod~2\pi [/Tex], then

[Tex] cot(x/2) = 2\sum_{j=1}^{\infty}\sin(jx) [/Tex] in the Cesáro sense.

In (i) I have tried to write out two geometric series and summing them, but I can't get the desired result. I have no idea on (ii).

"in the Cesáro sense" means (i think) that the average of a given sequence [Tex] s_0,s_1,s_2,\ldots [/Tex] converges against a given limit L (the sequence itself doesn't nescessarily) - that is, the sequence [Tex] s_0, (s_0 + s_1)/2, (s_0 + s_1 + s_2)/3,\ldots \rightarrow L [/Tex].