All Questions
11
questions
1
vote
1
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122
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Show that $\left | \sum_{k=1}^\infty \frac{\left(e^{ik} + \sin(\sqrt 2 k) + \cos(\sqrt 3 k)\right)^3}{k} \right| < 6$
I am trying to show the following sum is bounded:
$$ \sum_{k=1}^\infty \frac{\left(e^{ik} + \sin(\sqrt 2 k) + \cos(\sqrt 3 k)\right)^3}{k}$$
and to show that the magnitude
$$\left | \sum_{k=1}^\infty \...
- 972
-1
votes
1
answer
88
views
Tighter upper bound of $\sum\limits_{k=1}^\infty\frac{\tanh'(\cosh k)}{\cosh(\sin k)}$
How do I find tight upper bounds of $$S=\sum_{k=1}^{\infty}\frac{\tanh'(\cosh k)}{\cosh(\sin k)}$$ ? The derivative of $\tanh$ is $\text{sech}^2$, so using $$S=\sum_{k=1}^\infty\frac{1}{\cosh^2(\cosh ...
- 6,539
2
votes
1
answer
191
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Infinite Sum of an Inverse Trig Expression
I am attempting to find either a closed form for the following infinite sum, or failing that, the value $p$ for which the sum converges to $2\pi$ (somewhere around $0.82$?).
$$\sum_{i=1}^\infty \...
- 124
1
vote
1
answer
133
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Divergence of $\lim_{N\to +\infty}[ \int_{1}^{N+1} f(x) dx -\sum_{n=1}^{N}f(n)] $
Divergence of $\lim_{N\to +\infty}[ \int_{1}^{N+1} f(x) dx -\sum_{n=1}^{N}f(n)] $ where $f(x)= sin(log_e x)(\frac{1}{x^{a}}-\frac{1}{x^{1-a}})$,$ 0<a<1/2$
My try -
$\lim_{N\to +\infty}[ \int_{1}...
user826941
2
votes
0
answers
134
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How to determine whether the sum ${\sum}_{k=1}^{\infty} {\sin(2^k)\over k}$ converges?
I saw a question on quora asking whether or not the sum ${\sum}_{k=0}^{\infty}{sin(2^k)\over n}$ is convergent. My opinion, and that of the other answers, is that Dirichlet's test could be used with {...
- 233
3
votes
4
answers
70
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how do I show this :$\sum_{k=0}^{2n }\binom{2n}{k} \sin ((n-k)x)=0$ , for every real $x$ and for every integer $n$?
My attempt fails to show this formula $\sum_{k=0}^{2n }\binom{2n}{k} \sin ((n-k)x)=0$ which I have accrossed in my textbook, using induction proof, but I think by induction seems very hard, I want to ...
user517526
1
vote
4
answers
241
views
Determine convergence / divergence of $\sum \sin \frac{\pi}{n^2}$
Determine convergence / divergence of
$$\sum \sin \frac{\pi}{n^2}$$
let $a_n= \sin \frac{\pi}{n^2}$
I attempted the integral test but on the interval $[1, \sqrt{2})$ it is increasing and ...
- 453
1
vote
0
answers
213
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sum of exponential of cosine functions
I need to simplify this summation:
$$\sum_{i=1}^N\sum_{j=1}^N \exp(A^2 \cos(\omega t+\theta_i)\cos(\omega t+\theta_j))$$
where $\theta_i=\frac{2\pi}{N}i$.
Without the exponential function the terms ...
- 1,221
0
votes
1
answer
126
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Error term in an infinite series
Well, I've the following series:
$$\frac{1}{\left(1+\epsilon\cos\left(x\right)\right)^2}=\sum_{\text{k}=0}^\infty\epsilon^\text{k}\left(1+\text{k}\right)\cos^\text{k}\left(x\right)\tag1$$
This ...
- 28.7k
1
vote
0
answers
904
views
Convergence Test for a series involving trigonometric part
Determine whether the following function is convergent or divergent? If convergent, what does it converge towards.
$$f(x)=-2\sum _{ n=-\infty }^{ \infty }{ \left( \frac { \left( -1 \right) ^{ n } }{...
- 415
0
votes
1
answer
113
views
trigonometric summation
Taking into consideration the functions
$$\sum_{t=0}^{n} \sin{(\theta + t \phi)}=\frac{\sin({\frac{(n+1)\phi}2})\sin{(\theta+\frac{n \phi}2)}}{\sin{(\frac{\phi}2)}}$$
and
$$\sum_{t=0}^{n}\cos{(\...
- 217