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### Is the $\sum\sin(n)/n$ convergent or divergent? [duplicate]

Possible Duplicate: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ So, in my calculus class (one I'm teaching, not taking), ...
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### Sum inequality: $\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$ [duplicate]

I'm interested in finding an elementary proof for the following sum inequality: $$\sum_{k=1}^n \frac{\sin k}{k} \le \pi-1$$ If this inequality is easy to prove, then one may easily prove that the sum ...
820 views

### Give a demonstration that $\sum\limits_{n=1}^\infty\frac{\sin(n)}{n}$ converges. [duplicate]

Possible Duplicate: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ Is the sum of sin(n)/n convergent or divergent? Give a ...
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### Why $\sum^{\infty}_{n=1}\frac{\sin[n]}{n}=\frac{1}{2}(\pi-1)$? [duplicate]

Possible Duplicate: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ From Stewart, we cannot find a calculus 2 easy way to ...
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### Evaluate $\sum_{n=1}^{\infty} \frac{\sin \ n}{ n }$ using the fourier series

I am a beginner with Fourier series and I have to evaluate the sum $$\sum_{n =1}^{\infty}{\sin\left(n\right) \over n}$$ I don't know which function I have to take to evaluate the fourier series ......
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### Show the divergence of the series $\sum \frac{\sin^2n}{n}$ without Dirichlet's test

Show that the series $$\sum_{n\in\mathbb N} \frac{\sin^2n}{n}$$ is divergent. I know how to do this with the Dirichlet's test. But is there any other way to prove it? Thanks!
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### How to prove $\sum\limits_{n=1}^\infty\frac{\sin(n)}n=\frac{\pi-1}2$ using only real numbers.

I noticed that a lot of the time, people ask whether the following sum converges: $$\sum_{n=1}^\infty\frac{\sin(n)}n$$ Though I've never stopped to ask what it equaled. According to this other post,...
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### Proof that $\displaystyle\sum_{n=1}^{\infty}{(-1)^{n+1}\sin(n)\over{n}}={1\over2}$

While messing around with WolframAlpha, I came across this identity that ${\sin{1}\over{1}}-{\sin{2}\over{2}}+{\sin{3}\over{3}}-{\sin{4}\over{4}}+{\sin{5}\over{5}}\cdots={1\over{2}}$. One would ...
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### How can we determine the convergence or divergence of $\sum_{k=1}^{n} \frac{\sin(\sqrt{k})}{\sqrt{k}}$?

Could any one find if the series: $$\sum_{k=1}^{n} \frac{\sin(\sqrt{k})}{\sqrt{k}}$$ is divergent or convergent? I tried various techniques, but none of them worked (absolute convergence, ...
### the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational
Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...