All Questions
Tagged with summation trigonometry
424
questions
0
votes
1
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82
views
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
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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
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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:
$$\...
- 640
0
votes
1
answer
84
views
If $\sum_{r=1}^5 cos(rx)=5$ find the number solutions it has in $[0,2\pi]$
if $$\sum_{r=1}^5 \cos(rx)=5$$
then find the number of solutions it has in $[0,2\pi]$.
I've tried two different methods to find the solution(s), but both of which are proving to be very lengthy.
...
- 1,296
6
votes
2
answers
296
views
How do we prove that :$\tan^2(10)+\tan^2(50)+\tan^2(70) =9$
Prove : $\tan^2(10) + \tan^2(50)
+ \tan^2(70) =9$
my attempt
Let $\text{t} :=\tan(10)$
$$\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
214
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
389
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 ...
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