Question

Prove or disprove that $ \sum\limits_{k = 1 }^T f(k)=0 $ where $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin(\frac{n(n+1)(2n+1)}{6}x) $

score 658 · Answer 1 · 2026-04-17 09:46:59

658

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Prove or disprove that $ \sum\limits_{k = 1 }^T f(k)=0 $ where $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin(\frac{n(n+1)(2n+1)}{6}x) $

Published on 17 Apr 2026 - 9:46

score 58 · Answer 2 · 2026-04-09 21:45:38

58

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How to convert my equation (Exponential) to cot form?

Published on 09 Apr 2026 - 21:45

score 270 · Answer 3 · 2026-04-14 12:55:45

270

Views

Polar Curve Fitting

Published on 14 Apr 2026 - 12:55

score 95 · Answer 4 · 2026-04-16 17:05:46

Number of solutions $2^\frac{1}{\sin^2 x_2} \cdot 3^\frac{1}{\sin^2 x_3}\cdot\;\cdots\;\cdot n^\frac{1}{\sin^2 x_n} \leq n!$ with $x_i\in(0,4\pi)$

score 386 · Answer 5 · 2026-04-16 11:03:34

386

Views

Conjecture about $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin\left(P_r(n) \frac{a \pi}{b}\right) $

Published on 16 Apr 2026 - 11:03

score 314 · Answer 6 · 2026-04-16 07:20:55

314

Views

Extended conjecture for $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin\left(\sum\limits_{k = 1 }^\infty \frac{a_k \pi}{b_k}P_k(n) \right)$

Published on 16 Apr 2026 - 7:20

score 91 · Answer 7 · 2026-04-11 23:10:51

91

Views

Expansion of $x$ in powers of $u$

Published on 11 Apr 2026 - 23:10

score 32 · Answer 8 · 2026-04-17 04:49:05

32

Views

$\sum_{k=1}^n\sin kx=\frac{\cos\left(\frac{1}2x\right)-\cos\left(n+\frac{1}2\right)x}{2\sin(x/2)},x\neq0,\pm\pi,\pm 2\pi,\ldots$

Published on 17 Apr 2026 - 4:49

score 175 · Answer 9 · 2026-05-10 16:13:18

175

Views

Any suggestions on how to compute $\limsup |\cos n|^{n^2}$?

Published on 10 May 2026 - 16:13

score 46 · Answer 10 · 2026-04-12 16:37:43

46

Views

Summation of Finite trigonometrical series

Published on 12 Apr 2026 - 16:37

score 47 · Answer 11 · 2026-04-11 01:42:17

47

Views

Closed form for a conditional trigonometric series

Published on 11 Apr 2026 - 1:42

score 79 · Answer 12 · 2026-04-15 21:50:36

Identical equation $\sin(\frac{nx}{2^{n-1}})=\sin x\times \sin(x+\pi/n)\times \cdot\cdot\cdot \times \sin(x+\frac{(n-1)\pi}{n}) $

score 385 · Answer 13 · 2026-04-15 04:36:02

385

Views

Sum of binomial coefficients in Gould tables

Published on 15 Apr 2026 - 4:36

score 407 · Answer 14 · 2026-04-14 06:10:00

407

Views

Simplify $\sum_{k = 1}^n \tan(k) \tan(k - 1)$ by first proving $\tan(k)\tan(k - 1) = \frac{\tan(k) - \tan(k - 1)}{\tan(1)} - 1$

Published on 14 Apr 2026 - 6:10

score 134 · Answer 15 · 2026-04-17 06:11:16

134

Views

Probably this :$\sum_{n=2}^{\infty}(-1)^n\frac{\arctan{(1-2^n)}\log n \tan{(1-2^{-n})}}{n^3\sqrt{n}\log \log n }$ is Euler constant

Published on 17 Apr 2026 - 6:11

List Question

Prove or disprove that $ \sum\limits_{k = 1 }^T f(k)=0 $ where $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin(\frac{n(n+1)(2n+1)}{6}x) $

How to convert my equation (Exponential) to cot form?

Polar Curve Fitting

Number of solutions $2^\frac{1}{\sin^2 x_2} \cdot 3^\frac{1}{\sin^2 x_3}\cdot\;\cdots\;\cdot n^\frac{1}{\sin^2 x_n} \leq n!$ with $x_i\in(0,4\pi)$

Conjecture about $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin\left(P_r(n) \frac{a \pi}{b}\right) $

Extended conjecture for $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin\left(\sum\limits_{k = 1 }^\infty \frac{a_k \pi}{b_k}P_k(n) \right)$

Expansion of $x$ in powers of $u$

$\sum_{k=1}^n\sin kx=\frac{\cos\left(\frac{1}2x\right)-\cos\left(n+\frac{1}2\right)x}{2\sin(x/2)},x\neq0,\pm\pi,\pm 2\pi,\ldots$

Any suggestions on how to compute $\limsup |\cos n|^{n^2}$?

Summation of Finite trigonometrical series

Closed form for a conditional trigonometric series

Identical equation $\sin(\frac{nx}{2^{n-1}})=\sin x\times \sin(x+\pi/n)\times \cdot\cdot\cdot \times \sin(x+\frac{(n-1)\pi}{n}) $

Sum of binomial coefficients in Gould tables

Simplify $\sum_{k = 1}^n \tan(k) \tan(k - 1)$ by first proving $\tan(k)\tan(k - 1) = \frac{\tan(k) - \tan(k - 1)}{\tan(1)} - 1$

Probably this :$\sum_{n=2}^{\infty}(-1)^n\frac{\arctan{(1-2^n)}\log n \tan{(1-2^{-n})}}{n^3\sqrt{n}\log \log n }$ is Euler constant

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