All Questions
Tagged with summation trigonometry
424
questions
0
votes
1
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82
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How to prove that $\sin\left(\frac{\pi}{2n}\right)\sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n}\pi\right)=1$? [closed]
How to prove that
$$\sin\left(\frac{\pi}{2n}\right)\sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n}\pi\right)=1$$
?
- 305
1
vote
0
answers
58
views
Deduce that $\sum_{n=1}^\infty\frac{1}{1+n^2}=\frac{1}{2}(\pi\coth\pi-1)\quad$ [duplicate]
I am having problems showing that $$\sum_{n=1}^\infty\frac{1}{1+n^2}=\frac{1}{2}(\pi\coth\pi-1)\quad $$
Here's my attept to this point:
I tried to express each term using a partial fraction ...
- 402
0
votes
0
answers
32
views
$\sum_{k=1}^{2m+1}\cos\left(\frac{2k\pi-\operatorname{cos^{-1}}(x)}{2m+1}\right)^n$ - $n$th power of the root of a polynomial of odd degree
Context
I started with the following (very common) problem:
Given this polynomial $p(x)$, calculate the sum/the sum of the squares/of the cubes of the roots"
So I wanted to see if I could find ...
1
vote
1
answer
36
views
Summation of infinite cos series and determining theta
Question: In the figure,
$A_0A_1,A_2A_3,A_4A_3...$
are all perpendicular to $L_1$
$A_1A_2,A_3A_4,A_5A_6...$
are all perpendicular to $L_2$
If $A_0A_1=1$
And $A_0A_1+A_1A_2+A_2A_3+A_3A_4......\...
1
vote
0
answers
98
views
Restructuring Jacobi-Anger Expansion
In Jacobi-Anger expansion, $$e^{\iota z \sin(\theta)}$$ can be written as:
$$e^{\iota z \sin(\theta)} = \sum_{n=-\infty}^{\infty} J_n(z)e^{\iota n \theta}$$
where $J_n(z)$ is the Bessel function of ...
- 31
1
vote
1
answer
170
views
How to derive the sum $ \sum_{k=1}^{n-1} \frac{1}{\cosh^2\left(\frac{\pi k}{n}\right)}$
\begin{align*}
\sum_{k=1}^{n-1} \frac{1}{\cosh^2\left(\frac{\pi k}{n}\right)}\end{align*}
I tried to solve with mathematica that shows
Does anyone know how to derive this and does it is possible for ...
8
votes
1
answer
250
views
Compute $\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$
Question: Compute $$\sum^{2024}_{k=1} \frac{2023-2022 \cos \left(\frac{\pi(2k-1)}{2024} \right)}{2021-2020 \cos \left(\frac{\pi(2k-1)}{2024} \right)}$$
I began by rearranging the sum as follows:
$$\...
- 640
0
votes
1
answer
85
views
If $\sum_{r=1}^5 cos(rx)=5$ find the number solutions it has in $[0,2\pi]$
if $$\sum_{r=1}^5 \cos(rx)=5$$
then find the number of solutions it has in $[0,2\pi]$.
I've tried two different methods to find the solution(s), but both of which are proving to be very lengthy.
...
- 1,326
6
votes
2
answers
296
views
How do we prove that :$\tan^2(10)+\tan^2(50)+\tan^2(70) =9$
Prove : $\tan^2(10) + \tan^2(50)
+ \tan^2(70) =9$
my attempt
Let $\text{t} :=\tan(10)$
$$\tan^2(10) + \tan^2(50)
+ \tan^2(70) = \tan^2(10) + \tan^2(60-10)
+ \tan^2(60+10)=t^2 + \left({\frac{\sqrt{...
- 2,288
4
votes
0
answers
87
views
How deduce $\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ from $\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$?
I know that $\displaystyle\prod_{k=1}^{n-1} \sin \frac{k\pi}{n} =\frac{n}{2^{n-1}}$ for any integer $n \geq 1$ is true.
Now, suppose that $n$ is odd, how show
$$
\prod_{k=1}^{(n-1)/2} \sin^2 \frac{k\...
- 323
4
votes
0
answers
135
views
Simplify a summation in the solution of $\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x$
Context
I calculated this integral:
$$\begin{array}{l}
\displaystyle\int_{0}^{\infty}e^{-cx}x^{n}\arctan(ax)\mathrm{d}x=\\
\displaystyle\frac{n!}{c^{n+1}}\left\lbrace\sum_{k=0}^{n}\left[\text{Ci}\left(...
16
votes
2
answers
666
views
Simplifying $3S_1 + 2S_2 + 2S_3$, where $S_1=2\sum_{k=0}^n16^k\tan^4{2^kx}$, $S_2=4\sum_{k=0}^n16^k\tan^2{2^kx}$, $S_3=\sum_{k=0}^n16^k$
If $$S_1=2\sum_{k=0}^n 16^k \tan^4 {2^k x}
$$
$$S_2=4\sum_{k=0}^n 16^k \tan^2 {2^k x}
$$
$$S_3= \sum_{k=0}^n 16^k
$$
Find $(3S_1 + 2S_2 + 2S_3)$ as a function of $x$ and $n.$
In the expression asked ...
- 491
3
votes
1
answer
215
views
Sum with Binomial Coefficients and Sine; $S=\sum_{k=0}^n \binom{n}{k} \sin(kx)$
Sum with Binomial Coefficients
Let $n ∈ ℕ₀$ and $x ∈ ℝ$.
$$S=\sum_{k=0}^n \binom{n}{k} \sin(kx)$$
Simplify the sum to a polynomial in n.
I tried to use Euler's Formula and the Binomial Theorem, ...
- 77
10
votes
1
answer
390
views
Is there an identity for $\sum_{k=0}^{n-1}\csc(w+ k \frac{\pi}{n})\csc(x+ k \frac{\pi}{n})\csc(y+ k \frac{\pi}{n})\csc(z+ k \frac{\pi}{n})$?
What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$...
- 1,379
3
votes
1
answer
90
views
Closed form for $\sum_{n=1}^{\infty}\frac{(2\log\phi)^{2n+3}B_{2n}}{2n(2n+3)!}$
I need a closed form for the sum $$\sum_{n=1}^{\infty}\frac{(2\log\phi)^{2n+3}B_{2n}}{2n(2n+3)!} $$
where $\phi=\frac{1+\sqrt{5}}{2}$ is the golden ratio and $B_n$ are the Bernoulli numbers.
I tried ...
- 862
1
vote
1
answer
122
views
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
$$\left | \sum_{k=1}^\infty \...
- 972
-1
votes
1
answer
126
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Find the value $\sum_{n=a}^b\frac1{\sin (2^{n+3})}$ [closed]
Find the value of: $$\sum_{n=0}^{10}\frac1{\sin (2^{n+3})}$$
I'm stuck on this problem, can someone please help?
- 596
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1
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58
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How to apply $\prod _{k=1}^n \cos (\theta _k)=\frac{1}{2^n}\sum _{\text{e$\epsilon $S}} \text{cos}(e_1 \theta _1+\text{...}+e_n \theta _n)$?
It is shown that the product-to-sum identities are given by:
$\prod _{k=1}^n \cos \left(\theta _k\right)=\frac{1}{2^n}\sum _{\text{e$\epsilon $S}} \text{cos}\left(e_1 \theta _1+\text{...}+e_n \theta ...
- 1,379
7
votes
2
answers
288
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Is there an identity for $\sum_{k=0}^{n-1}\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)$?
What I'd like to find is an identity for $$\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
0
votes
1
answer
77
views
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 $\...
- 3
2
votes
3
answers
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
votes
3
answers
157
views
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
-1
votes
1
answer
88
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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 $$S=\sum_{k=1}^\infty\frac{1}{\cosh^2(\cosh ...
- 6,539
4
votes
1
answer
131
views
Prove that $\sum_{l=0}^{N-1}\frac{\sin^2(\pi x)}{\sin^2(\frac{\pi}{N}(l-k+x))}=N^2$ [closed]
When I numerically compute the sum below it is always $1$. How can I prove this? $N$ is an integer number and $k$ is an integer number between $0$ to $N-1$ and $x$ is real number between $0$ and $0.5$
...
- 369
7
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3
answers
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Proving $\sum_{k=1}^{2n-1}\frac{\sin(\frac{\pi k^2}{2n})}{\sin(\frac{\pi k}{2n})}=n$
I wander on the internet and found this problem (from Quora) this link
The problem is proving the identity: $$\sum_{k=1}^{2n-1}\frac{\sin\left(\frac{\pi k^2}{2n}\right)}{\sin\left(\frac{\pi k}{2n}\...
- 2,702
1
vote
1
answer
159
views
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
vote
1
answer
120
views
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 $$S=\sum_{n=1}^{\infty}\left(\left(\...
0
votes
0
answers
82
views
Calculate Min/Max of sum of absolute sines
I need to calculate as part of a proof the maximum and minimum of this function analytically:
$$f_n(\varphi) = \sum_{k=0}^{n-1}\left|\sin \left(\varphi-\frac{2\pi k}{n}\right) \right|$$
whereby $\...
- 133
3
votes
0
answers
107
views
Series representation of $n$th derivative of $x^n/(1+x^2)$
Find the nth derivative of $\frac{x^n}{1+x^2}$.
Please I need help in this. They are further asking to show that when $x=\cot y$ the nth derivative can be expressed as
$$n!\sin y\sum_{r=0}^{n}(-1)^r {...
2
votes
0
answers
111
views
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
$$x_{kl}=\frac{1}{4}\Big(\frac{1}{\sin\alpha_{kl}}+\frac{...
- 1,998
2
votes
0
answers
97
views
Integral of $\,\tan^{2n}(x)\,\mathrm dx$
I want to evaluate $\,\displaystyle I_{n}=\int_{0}^{\frac{\pi}{4}} \tan^{2n}(x)\,\mathrm dx$.
I proved that $\,I_{n}+I_{n-1}=\dfrac{1}{2n-1}\,,\,$ where $I_{0}=\dfrac{\pi}{4}$.
From that I found that (...
user1009942
2
votes
2
answers
247
views
How to prove identity $\sum_{k=0}^{n-1}\tan^2\left(\theta+{k\pi\over n}\right)=n^2\cot^2\left({n\pi\over2}+n\theta\right)+n(n-1)$?
Looking at Jolley, Summation of Series, formula 445:
$\sum_{k=0}^{n-1}\tan^2\left(\theta+{k\pi\over n}\right)=n^2\cot^2\left({n\pi\over2}+n\theta\right)+n(n-1)$
How can one prove this?
Considering $\...
- 1,379
1
vote
1
answer
50
views
Find minimum n :$ \prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{\sqrt{27k^3+54k^2+36k+8}}\right)}{\arctan \left( \frac{1}{\sqrt{3k+1}} \right)}>2000$ [closed]
We've got to find the minimum value of n for which
$$ \prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{\sqrt{27k^3+54k^2+36k+8}}\right)}{\arctan \left( \frac{1}{\sqrt{3k+1}} \right)}>2000$$
So I just ...
- 157
10
votes
1
answer
304
views
Is there an identity for $\sum_{k=0}^{n-1}\tan^4\left({k\pi\over n}\right)$?
Is there a simple relation for $\sum_{k=0}^{n-1}\tan^4\left({k\pi\over n}\right)$
like there is for $\sum_{k=0}^{n-1}\tan^2\left({k\pi\over n}\right)$?
Looking at Jolley, Summation of Series, formula ...
- 1,379
18
votes
2
answers
496
views
Prove $\sum^{n+1}_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \frac n4$
How do you prove that $\displaystyle\sum^{n+1}_{j=1}\left|\cos\left(j\cdot x\right)\right|\geqslant \dfrac{n}{4}$, where $x\in\mathbb{R}$?
I tried mathematical induction, but it doesn't work. I also ...
- 405
1
vote
0
answers
49
views
Sum of inverse tangents of the roots of a polynomial in terms of its coefficients
I posted a similar question to this, however this question refers to the sum of inverse tangents of any polynomial.
Let $$P(x)=n_1x^{k}+n_2x^{k-1}+n_3x^{k-2}+...+n_{k-1}x+n_{k}$$
where $k$ is a ...
- 977
0
votes
5
answers
199
views
Evaluating $\sum_{k=1}^{10}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right)$
Find the value of $$\sum_{k=1}^{10}\left(\sin\frac{2k\pi}{11}+i\cos\frac{2k\pi}{11}\right)$$
I have heard somewhere that this question can be done by $nth$ roots of unity or by vector algebra. But I'...
- 1,253
1
vote
2
answers
66
views
Closed form for $\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x)$
I request a closed form for the following sum $$S(n)=\sum_{r=0}^{n}(-1)^r {3n+1 \choose 3r+1} \cos ^{3n-3r}(x)\sin^{3r+1} (x) $$
I tried using De Moivre's theorem $$\cos(nx)+i\sin(nx)=(\cos (x)+i\sin(...
user1103841
2
votes
1
answer
62
views
If $\alpha+\beta+\gamma=\pi$, then does $\sqrt{\sin\alpha}+\sqrt{\sin\beta}+\sqrt{\sin\gamma}$ reach its maximum when $\alpha=\beta=\gamma$? [closed]
In any given triangle ($\alpha+\beta+\gamma= \pi$), the following inequality holds:
$$\sin {\alpha}+\sin{\beta}+ \sin{\gamma} \leq \frac{3 \sqrt{3}}{2}$$
with the maximum value of $\frac{3 \sqrt{3}}{2}...
- 51
0
votes
0
answers
51
views
Conjecture: $\sum_{n=1}^N\left|1+\frac{1+\tan(n)}{n\left(1+\tan(n+1)\right)}\right|-N-\ln^2N<0$, for sufficiently-large $N$
It's a follow up of Do we have $\prod_{n=1}^{\infty}\left(1+\frac{\tan\left(n\right)+1}{n\left(\tan\left(n+1\right)+1\right)}\right)=^?0$
Conjecture/Problem:
It seems we have for $N$ sufficiently ...
0
votes
0
answers
45
views
Evaluate $\sum_{n=1}^\infty \dfrac{\sin n}n$ [duplicate]
Evaluate $\sum_{n=1}^\infty \dfrac{\sin n}n$.
By Euler's theorem, $\sin x = \dfrac{e^{ix}-e^{-ix}}{2i}$ for every real number x. Also, I know that $\sum_{n=1}^\infty \dfrac{1}{n^2} = \dfrac{\pi^2}6.$ ...
- 1,837
1
vote
2
answers
77
views
computation of $\sum_{k=0}^{2p-1}( \cos(x+\frac{k \pi}{2p}))^{2p}$
Let $p \in \mathbb{N^*}, x \in \mathbb{R}$, someone I know was trying to compute the following sum:
$$\sum_{k=0}^{2p-1}\left( \cos\left(x+\frac{k \pi}{2p}\right)\right)^{2p}$$
It seems that the result ...
- 1,928
3
votes
1
answer
62
views
$\displaystyle P(n):|\sum_{k=1}^{n} \sin(k)\sin(k^2)| \leq 1, \forall n\geq 1$
Prove that $\displaystyle P(n):|\sum_{k=1}^{n} \sin(k) \sin(k^2)| \leq 1, \forall n\geq 1$ is true
What I've tried:
For $n=1 \implies \left|\sin(1)\sin(1^2)\right| \leq1$ is true.
Suppose that $P(n)$...
- 996
-3
votes
1
answer
73
views
how can we prove that sum of cosine can be sine function [closed]
I got this equation in a research paper but I couldn't understand how it was calculated:
$$\sum_{n=0}^{N-1}\text{cos}\left[ \frac{(\pi k/2)\left( 2n+1 \right)}{2N} \right]=\frac 12\frac{\text{sin}\...
- 55
2
votes
1
answer
94
views
Prove that for integer m and N, this sum with N-1 terms of cosec raised to 2m multiplied by (N/2)^m is an integer.
Prove that for integers $m \ge 1$, $N \ge 2$,
$F(m,N)=\large \frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\frac{\pi j}{N}\right)$ is an integer.
I encountered this ...
- 838
3
votes
1
answer
125
views
Find the maximum value of $f(0)$ where $f$ ranges over all elements of $C$.
Let $C$ be the set of functions $f(x) = 1 + \sum_{n=1}^N a_n\cos(2\pi nx),$ where $f(x)\ge 0$ for all $x\in \mathbb{R}$, $a_n = 0$ when $n\equiv 0\mod 3$, and each $a_n\in \mathbb{Z}.$ Find the ...
- 1,137
2
votes
1
answer
191
views
Infinite Sum of an Inverse Trig Expression
I am attempting to find either a closed form for the following infinite sum, or failing that, the value $p$ for which the sum converges to $2\pi$ (somewhere around $0.82$?).
$$\sum_{i=1}^\infty \...
- 124
1
vote
1
answer
55
views
If $A+B+C+D=\pi$ then find $\sum\cos A\cos B-\sum\sin A\sin B$
Question:
If $A+B+C+D=\pi$ then find $\sum\cos A\cos B-\sum\sin A\sin B$
My Attempt:
$$\cos((A+B)+(C+D))=-1\\\implies\cos(A+B)\cos(C+D)-\sin(A+B)\sin(C+D)=-1\\\implies(\cos A\cos B-\sin A\sin B)(\cos ...
- 8,318
-2
votes
1
answer
100
views
How to find the approximate value of $\operatorname{arcosh}$?
Does anyone knows a good way to approximate $\operatorname{arcosh}$ between $1.0$ and $1.1$ precisely? Me and some others are using the standard series $$\ln(2x)-\sum_{n=1}^\infty\left(\frac{(2n)!}{2^{...
- 203
0
votes
1
answer
98
views
upper bounds for partial sums of some cosine series
I was wondering how one could show the following results:
$|\sum_{n=1}^N (-1)^n \cos^2 (n+1)| \leq 1$ for all $N\ge 1$.
There exists a real number $M$ so that $|\sum_{n=1}^N (-1)^n \cos(2(n+1))|\leq ...
- 1,166