Linked Questions

18
votes
1answer
20k views

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), ...
15
votes
2answers
2k views

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 ...
2
votes
1answer
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 ...
0
votes
1answer
261 views

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 ...
0
votes
0answers
213 views

Evaluate $\sum_{k=1}^{\infty}\frac{\sin k}{k}$. [duplicate]

Possible Duplicate: Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ Evaluate $\displaystyle\sum_{k=1}^{\infty}\frac{\sin k}{...
2
votes
1answer
103 views

Problem dealing with $\sum \frac{\sin(n)}{n}$ and its convergence [duplicate]

$$\text{If} \ S=\displaystyle\sum_{n=1}^{\infty}\dfrac{\sin (n)}{n}, \ \text{then what is} \ 2S+1$$ I know that $\sum \frac{\sin(n)}{n}$ converges. But now what do I do?
10
votes
3answers
14k views

Does $\sum\frac{\sin n}{n}$ converge? [duplicate]

Does $\sum\frac{\sin n}{n}$ converge? I have tried the comparison test, root test and ratio test but still can't prove it is convergent or divergent.
21
votes
1answer
1k views

How to prove $\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\frac{\pi-1}{2}$

One of my classmates challenged me to solve $\displaystyle\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\;?$ With a simple c program I found that $\displaystyle\sum\limits_{n=1}^{1048576}\frac{\sin n}n\...
8
votes
4answers
1k views

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 ......
1
vote
3answers
308 views

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!
8
votes
1answer
335 views

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,...
4
votes
2answers
91 views

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 ...
2
votes
1answer
102 views

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, ...
2
votes
1answer
138 views

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 ...
1
vote
0answers
113 views

Computation of the series: $\sum_{n=1}^{\infty} \frac{\sin (n)}{n}$ [duplicate]

How to compute the series above? Thanks in advance.

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