The trigonometric-series tag has no wiki summary.
4
votes
2answers
39 views
Looking for a source of an infinite trigonometric summation and other such examples.
Question:
If $x \neq 0$, then prove that $\displaystyle \sum_{n=1}^{\infty}\dfrac1{2^n} \tan\left(\dfrac{x}{2^n}\right) = \dfrac1{x} - \cot x.$
My answer:
I proved this result by using the ...
18
votes
1answer
146 views
Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$
Please help me to find a closed form for the infinite product
$$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$
where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
6
votes
1answer
83 views
Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$
I need help with calculating this sum:
$$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
8
votes
1answer
62 views
Closed form for $\sum_{n=1}^\infty\frac{\cos(\pi \log n)}{n^2}$
Is there a closed form for the following sum? $$\sum_{n=1}^\infty\frac{\cos(\pi\log n)}{n^2}$$
1
vote
0answers
22 views
Understanding the indices in a Fourier series
Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written
$$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$
which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
1
vote
2answers
65 views
$\cot(x)\,$ in the large $x$ limit?
I couldn't find asymptotic forms of trigonometric functions in any Math Table.
In particular, I am trying to find $\;\cot(a x)\;$ in large $x$ limit.
thanks,
2
votes
1answer
44 views
Sum of tangent functions where arguments are in specific arithmetic series
By looking through an book, I found this interesting series
To prove that:
$$\tan(\theta)+\tan \left(\theta+ \frac{\pi}{n} \right) + \tan(\theta + \frac{2\pi}{n}) + \dots + \tan \left (\theta + ...
1
vote
2answers
92 views
Algebraic solution to find circle radius given distance of three external points from perimeter
I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is ...
5
votes
1answer
66 views
finding $\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$
I want to find
$$L=\lim_{n\rightarrow\infty}\sum_{r=1}^{n}\arctan\left(\frac{2r}{1-r^2+r^4}\right)$$
I already know that I need to split the expression $\frac{2r}{1-r^2+r^4}$ of the form ...
2
votes
2answers
159 views
Power Reducing Formula
I need some how with the power reducing formula. I'm having trouble understanding how to apply it. Here's an equation that utilizes it:
$\cos^2 (\theta x) - 1 = 0$
How do I solve this on the ...
2
votes
3answers
162 views
Find sum of the Trignomertric series
Q1: The sum of the infinite series $\cot ^{-1}2 + \cot ^{-1} 8+ \cot^{-1}18+ \cot^{-1}32\cdots$
1.$\pi/3$
2.$\pi/4$
3.$\pi/2$
4.None
Q2: Value of $\lim_ {n \to \infty}[ {\cos \frac{\pi}{2^2} } ...
1
vote
1answer
70 views
Trigonometry Identities
Consider a collection of five points evenly spaced around a circle to form a regular
pentagon. Assume the figure is scaled so that the sides of the pentagon have length 1.
Question: Use Ptolemy’s ...
0
votes
1answer
53 views
trigonometric summation
Taking into consideration the functions
$$\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)}}$$
and
...
4
votes
0answers
85 views
Has this function a name?
Has this function a name?
$$f(x) = \prod_{i=2}^{\lceil \frac{x}{2} \rceil} \sin\left( \frac {\pi x}{i}\right)$$
(the product of $\sin( \frac {\pi x}{i})$ for $i=2,3,...,\lceil x/2 \rceil$)
2
votes
0answers
67 views
On the series $ \displaystyle \sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) $.
Is the following series ‘summable’ in the sense that it may be divergent but we can attach a meaning to it?
$$
\sum_{n=1}^{\infty} \frac{\Lambda(n)}{\sqrt{n}} \cos(x \log(n) + a) = (\text{reg}) ...
6
votes
1answer
189 views
Does this curve tend to a square wave?
I have put some Mathematica code here:
http://pastebin.com/cY6r7skS
that uses this algorithm:
$$y1 = Sin[x];$$
$$y2 = Sin[y1];$$
$$y3 = Sin[y1 + y2];$$
$$y4 = Sin[y1 + y2 + y3];$$
$$y5 = Sin[y1 + y2 ...
1
vote
1answer
64 views
Confused as to the right answer to this summation, am I wrong (most likely) or is the answer provided wrong?
If you have $\sum_{n = 0}^\infty(4/5)^n$ and you are asked to represent it as a geometric series you would:
$\sum_{n = 0}^\infty(4/5)(4/5)^{n-1}$ //factor out your constant
therefore $a = 4/5$, ...
0
votes
0answers
99 views
Zeroes of real trigonometric polynomials
Given the real trigonometric polynomial
$$
p(x)=\sum_{n=-N}^N c_n {\rm e}^{\,j n x}
$$
where $c_n=c_{-n}^{\ast}$, I'm wondering if there are methods to compute the zeros of $p(x)$ that take into ...
2
votes
2answers
118 views
Evaluation of a trigonometric partial sum
I just wanted to evaluate
$$ \sum_{k=0}^n \cos k\theta $$
and I know that it should give
$$ \cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)} $$
...
0
votes
0answers
85 views
Cosine series as a Fourier series.
Theorem:
Let $a_{n}\downarrow 0$ and suppose that $\left(a_{n}\right)$ is a
quasi-convex. Then $\displaystyle \frac{a_{0}}{2}+\sum a_{n}\cos nx$ is the Fourier series of the $L^{1}$ function ...
6
votes
2answers
174 views
Calculate: $\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$
Calculate the following sum for integers $n\ge2$:
$$\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$$
I'm trying to obtain a closed form if that is possible.
0
votes
0answers
83 views
Trigonometric series as a Fourier series of essentially bounded function.
A trigonometric series $ \displaystyle \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_{n}\cos nx +b_{n}\sin nx)$ is a Fourier series of a essentially bounded function if and only if there exists a constant $K$ ...
