Question

If a trigonometric series equals zero at every point, will its coefficients all be zero?

score 87 · Answer 1 · 2023-03-31 09:05:41

87

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If a trigonometric series equals zero at every point, will its coefficients all be zero?

Published on 31 Mar 2023 - 9:05

score 132 · Answer 2 · 2023-04-04 13:17:41

132

Views

Prove a relation based on angles $\frac{\pi}{11}$ and $\frac{\pi}{22}$

Published on 04 Apr 2023 - 13:17

score 35 · Answer 3 · 2023-04-06 03:35:42

What does $f(x)=\lim_{\ell \to \infty} \ell\sin(\frac{x}{\ell})-\sum_{n=1}^{\ell-1}\frac{n\sin(\frac{x}{n})}{\ell-1}$ converge to, if at all?

score 152 · Answer 4 · 2023-04-06 05:18:01

152

Views

A series in cosine : $\sum_{j=1}^{n} (-1)^{j-1}\cos\left(\frac{j\pi}{2n+1}\right)$

Published on 06 Apr 2023 - 5:18

score 263 · Answer 5 · 2023-04-07 06:48:12

263

Views

Are sums ${ \sum _{j=1} ^{n-1} \frac{1}{\sin( \frac{2 \pi}{2n} j) ^{2m} } }$ rational?

Published on 07 Apr 2023 - 6:48

score 42 · Answer 6 · 2026-03-27 11:45:23

42

Views

Approximation degree by trigonometric polynomial

Published on 27 Mar 2026 - 11:45

score 69 · Answer 7 · 2023-05-04 14:46:11

69

Views

$\frac{1}{7\operatorname{cosec}^{2}(7\theta)}{\sum_{k=0}^{6} \operatorname{cosec}^{2}(\theta+\frac{k\pi}{7})}$

Published on 04 May 2023 - 14:46

score 39 · Answer 8 · 2023-05-04 15:51:01

39

Views

Solving trigonometric series of $cosec^{2}$

Published on 04 May 2023 - 15:51

score 167 · Answer 9 · 2023-05-06 12:54:09

167

Views

What is the value of $\sum_{k=1}^{\infty}\frac{\cos\left(\frac{2\pi k}{3}\right)}{k^2}$?

Published on 06 May 2023 - 12:54

score 76 · Answer 10 · 2023-05-07 09:09:35

76

Views

Analytically showing that $\sum_{k=1}^N \frac{\sin^2\phi_k}{N(a^2+b^2-2ab\cos\phi_k)^2}$ is independent of $N$ for large $N$

Published on 07 May 2023 - 9:09

score 67 · Answer 11 · 2023-05-07 20:07:28

67

Views

Closed form for the finite sum $\sum_{h=1}^{\frac{k-1}{2}} \tan\frac{h\pi n}{k}$

Published on 07 May 2023 - 20:07

score 58 · Answer 12 · 2023-05-08 09:51:51

58

Views

On proving that the sum $\sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational using trigonometry.

Published on 08 May 2023 - 9:51

score 265 · Answer 13 · 2026-03-25 20:53:13

265

Views

Is there an identity for $\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$?

Published on 25 Mar 2026 - 20:53

score 47 · Answer 14 · 2023-05-13 20:21:25

47

Views

How to apply $\prod _{k=1}^n \cos (\theta _k)=\frac{1}{2^n}\sum _{\text{e$\epsilon $S}} \text{cos}(e_1 \theta _1+\text{...}+e_n \theta _n)$?

Published on 13 May 2023 - 20:21

score 34 · Answer 15 · 2023-06-02 14:45:58

34

Views

On Stein&Shakarchi's discussion on the "uniqueness of Fourier series"

Published on 02 Jun 2023 - 14:45

List Question

If a trigonometric series equals zero at every point, will its coefficients all be zero?

Prove a relation based on angles $\frac{\pi}{11}$ and $\frac{\pi}{22}$

What does $f(x)=\lim_{\ell \to \infty} \ell\sin(\frac{x}{\ell})-\sum_{n=1}^{\ell-1}\frac{n\sin(\frac{x}{n})}{\ell-1}$ converge to, if at all?

A series in cosine : $\sum_{j=1}^{n} (-1)^{j-1}\cos\left(\frac{j\pi}{2n+1}\right)$

Are sums ${ \sum _{j=1} ^{n-1} \frac{1}{\sin( \frac{2 \pi}{2n} j) ^{2m} } }$ rational?

Approximation degree by trigonometric polynomial

$\frac{1}{7\operatorname{cosec}^{2}(7\theta)}{\sum_{k=0}^{6} \operatorname{cosec}^{2}(\theta+\frac{k\pi}{7})}$

Solving trigonometric series of $cosec^{2}$

What is the value of $\sum_{k=1}^{\infty}\frac{\cos\left(\frac{2\pi k}{3}\right)}{k^2}$?

Analytically showing that $\sum_{k=1}^N \frac{\sin^2\phi_k}{N(a^2+b^2-2ab\cos\phi_k)^2}$ is independent of $N$ for large $N$

Closed form for the finite sum $\sum_{h=1}^{\frac{k-1}{2}} \tan\frac{h\pi n}{k}$

On proving that the sum $\sqrt{1001^2 + 1}+\sqrt{1002^2 + 1} \ + ... + \sqrt{2000^2 + 1}$ is irrational using trigonometry.

Is there an identity for $\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$?

How to apply $\prod _{k=1}^n \cos (\theta _k)=\frac{1}{2^n}\sum _{\text{e$\epsilon $S}} \text{cos}(e_1 \theta _1+\text{...}+e_n \theta _n)$?

On Stein&Shakarchi's discussion on the "uniqueness of Fourier series"

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