# Tagged Questions

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### Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
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### Does the sequence $\{\sin^n(n)\}$ converge?

Does the sequence $\{\sin^n(n)\}$ converge? Does the series $\sum\limits_{n=1}^\infty \sin^n(n)$ converge?
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### Does $\sqrt[n]{\left\lvert \sin(2^n) \right\rvert}$ have a limit?

The entire question is essentially given in the title: For $n$ a positive integer, does the sequence $x_n = \sqrt[n]{\left\lvert \sin(2^n) \right\rvert}$ have a limit as $n \to \infty$? ...
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### Least upper bound for a positive real sequence satisfying $|x_u−x_v|\cdot|u−v|>1$

The starting point of this question is the: IMO 1991, problem 6: Prove that for any $\alpha>1$, there exist a bounded real sequence $\{x_n\}_{n\in\mathbb{N}}$ such that, if $u,v$ are distinct ...
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### For what $\lambda$ does the sequence $\left\{ \frac{k}{2\lambda} \right\}$ converge?

I am reading some materials on hypergeometric functions and their relation to general L-functions and the authors make use of the following "well-known elementary result from diophantine ...