# Tagged Questions

For questions about or related to trigonometric series.

326 views

### 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 ...
115 views

137 views

### 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 ...
84 views

### 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 ...
79 views

154 views

### 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 ...
34 views

### A geometric proof for the “small angle approximation” for the sine, cosine and tangent

How can be deduced the so called "small angle approximations" for sine, cosine and tangent, namely $\sin \theta \approx \theta$ $\tan\theta \approx \theta$ $\cos\theta \approx 1-\frac{\theta^2}{2}$ ...
52 views

30 views

21 views

### 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 ...