All Questions
Tagged with summation trigonometry
424
questions
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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
$$\left | \sum_{k=1}^\infty \...
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1
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126
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Find the value $\sum_{n=a}^b\frac1{\sin (2^{n+3})}$ [closed]
Find the value of: $$\sum_{n=0}^{10}\frac1{\sin (2^{n+3})}$$
I'm stuck on this problem, can someone please help?
- 596
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1
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How to apply $\prod _{k=1}^n \cos (\theta _k)=\frac{1}{2^n}\sum _{\text{e$\epsilon $S}} \text{cos}(e_1 \theta _1+\text{...}+e_n \theta _n)$?
It is shown that the product-to-sum identities are given by:
$\prod _{k=1}^n \cos \left(\theta _k\right)=\frac{1}{2^n}\sum _{\text{e$\epsilon $S}} \text{cos}\left(e_1 \theta _1+\text{...}+e_n \theta ...
- 1,379
7
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2
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Is there an identity for $\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$?
What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$$
here it can be shown that where $x=y$,
$$n^2 \csc^2(nx) = \sum_{k=0}^...
- 1,379
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1
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77
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Analytically showing that $\sum_{k=1}^N \frac{\sin^2\phi_k}{N(a^2+b^2-2ab\cos\phi_k)^2}$ is independent of $N$ for large $N$
Numerically, I have found that the following formula seems to be independent of $N$ for any choice of $a$ and $b$ at large $N$:
$$\sum_{k=1}^N \frac{\sin^2\phi_k}{N(a^2+b^2-2ab\cos\phi_k)^2}$$
with $\...
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2
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3
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173
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What is the value of $\sum_{k=1}^{\infty}\frac{\cos\left(\frac{2\pi k}{3}\right)}{k^2}$?
I've come across the following trigonometric series:
$$\sum_{k=1}^{\infty}\frac{\cos\left(\frac{2\pi k}{3}\right)}{k^2}$$
for which WolframAlpha gives the answer $-\dfrac{\pi^2}{18}$.
How do you ...
- 359
7
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3
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157
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History of the general formula for linearising $\cos^n(x)$
I was wondering where the formula:
$$\cos^n(x)=\sum_{k=0}^n\frac{n!}{k!(n-k)!}\cos(x(2k-n))$$
Was first originally published. I accidentally derived it a few days ago and was wondering where it was ...
- 474
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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 $$S=\sum_{k=1}^\infty\frac{1}{\cosh^2(\cosh ...
- 6,539
4
votes
1
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131
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Prove that $\sum_{l=0}^{N-1}\frac{\sin^2(\pi x)}{\sin^2(\frac{\pi}{N}(l-k+x))}=N^2$ [closed]
When I numerically compute the sum below it is always $1$. How can I prove this? $N$ is an integer number and $k$ is an integer number between $0$ to $N-1$ and $x$ is real number between $0$ and $0.5$
...
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7
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3
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Proving $\sum_{k=1}^{2n-1}\frac{\sin(\frac{\pi k^2}{2n})}{\sin(\frac{\pi k}{2n})}=n$
I wander on the internet and found this problem (from Quora) this link
The problem is proving the identity: $$\sum_{k=1}^{2n-1}\frac{\sin\left(\frac{\pi k^2}{2n}\right)}{\sin\left(\frac{\pi k}{2n}\...
- 2,702
1
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1
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159
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Special sum to show by hand : $S<e$
Problem :
Show that :
$$S=\sum_{n=1}^{104}\arcsin\left(\frac{2}{n^2+1}\right)<e$$
Without a computer (by hand).
This problem seems very difficult.
To show it I have used Jordan's inequality :
Let $...
1
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1
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120
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Find the value of sum $\forall\:\:\alpha,\beta\in\mathbb{R}$
Evaluate the sum $\forall\:\:\alpha,\beta\in\mathbb{R}$ $$S=\sum_{n=1}^{\infty}\frac{\alpha^{n+1}-1}{n(n+1)}\sin\left(\frac{n\pi}{\beta}\right)$$
I rewrote this as $$S=\sum_{n=1}^{\infty}\left(\left(\...
0
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0
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82
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Calculate Min/Max of sum of absolute sines
I need to calculate as part of a proof the maximum and minimum of this function analytically:
$$f_n(\varphi) = \sum_{k=0}^{n-1}\left|\sin \left(\varphi-\frac{2\pi k}{n}\right) \right|$$
whereby $\...
- 133
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Series representation of $n$th derivative of $x^n/(1+x^2)$
Find the nth derivative of $\frac{x^n}{1+x^2}$.
Please I need help in this. They are further asking to show that when $x=\cot y$ the nth derivative can be expressed as
$$n!\sin y\sum_{r=0}^{n}(-1)^r {...
2
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111
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A trigonometric sum [closed]
For $k=0,\cdots, m$ and $l=0,\cdots 2m+1$ let us put
$$ \alpha_{kl}=\frac{4k-4l+1}{4(m+1)}\pi\quad \beta_{kl}=\frac{4k+4l+3}{4(m+1)}\pi $$
and
$$x_{kl}=\frac{1}{4}\Big(\frac{1}{\sin\alpha_{kl}}+\frac{...
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