# Tagged Questions

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### Find $\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$

Find the following limit: $$\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$$ The numerator can be simplified by using Euler's formula and the sum of ...
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### Prove $\cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1$

$$\cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1$$ Wolframalpha shows that it is a correct identity, ...
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### Convergence of $\sum\frac{\tan(nz)}{n^2}$ to an analytic function…what if $z\in \mathbb{R}$?

For which values of $z$ does $$\sum_{n=1}^\infty \frac{\tan(nz)}{n^2}$$ converge? For which values of $z$ is the limiting function analytic? One can show, as in this answer, that ...
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### How does a taylor series of a binomial function equals a trigonometric function? [closed]

Any proof or derivation for the sinx and cosx function would be help. Image taken from http://en.wikipedia.org/wiki/Taylor_series
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### Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|<2\sqrt{\pi}$

For any real $x$ and positive integer $n$, is it true that: $$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$ Please justify.
### Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$
Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0