# Tagged Questions

For questions about or related to trigonometric series.

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### Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
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### Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right)$$ I know that without the power one could ...
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### Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
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### Proving that sin(x)/x=(1-x^2/pi^2)(1-x^2/4pi^2)(…)

I am faced with explaining to a bunch of people who have taken a course in real analysis, but no course in complex analysis, why $\sin(x)=x\prod_{n\geq1}(1-x^2/\pi^2n^2)$. I vaguely remember being ...
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### How to simplify summation containing trignometric function?

Prove that $$2 \sum\limits_{n=1}^\infty \left(\frac{x}{b}\right)^n \frac{\sin A}{n} = \tan^{-1}\left(\frac{2bx \cos A}{b^2-x^2}\right)$$ I have tried to do by changing this into imaginary part of ...
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### Trigonometric series as a Fourier series of essentially bounded function.

A trigonometric series $\displaystyle \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx +b_{n}\sin nx)$ is a Fourier series of a essentially bounded function if and only if there exists a constant $K$ ...
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### Where were my mistakes when I've combined $\eta(s)=\left(1-2^{1-s}\right)\zeta(s)$ and the Fourier series for the fractional part?

Let $s=x+it$ the complex variable, thus we are denoting $\Re s=x$. Combining the identity $$\eta(s)=\left(1-2^{1-s}\right)\zeta(s),$$ that holds for $0<x<1$, where $\zeta(s)$ is the Riemann Zeta ...