# Tagged Questions

For questions about or related to trigonometric series.

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### How to find the sum $\cos(2x)+\cos(4x)+\cos(6x)+\cos(8x)+\ldots+\cos(2016x)$? [duplicate]

Find the sum $$M= \cos(2x)+\cos(4x)+\cos(6x)+\cos(8x)+\ldots+\cos(2016x)$$
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### Using a taylor polynomial to approximate $cos(\frac14)$ with an error no more than $10^{-12}$

So the lagrange remainder is given by: $$R_n(x)=\frac{f^{n+1}(c)}{(n+1)!}(x-a)^{n+1}$$ We want $cos(\frac14)$ and we can do it around a=0. We know that $f^{n+1}$ is either $\pm$ cosx or $\pm$sinx, ...
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### Trigonometric series convergence

I was interested in evaluating $$\sum_{n=1}^{\infty} \frac{\cos n}{n+k}$$ I saw with computation that many of such series converge. Is there a general result? I've tried using Taylor expansion of ...
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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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### Product of trigonometric polynomials is a trigonometric polynomial

A trigonometric polynomial was defined as $$f(x) = \frac{a_0}{2} + \sum_{k=1}^{n}(a_k \cos(kx) + b_k \sin(kx))$$ I heard somewhere that trigonometric polynomials have a ring structure, i.e. a product ...
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### Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right)$$ ...
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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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