For questions about or related to trigonometric series.

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1answer
34 views

Is $H(\theta) = \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos (2\pi n_k \theta)$ for a given sequence $n_k$ equal a.e. to a continuous function?

I am studying Furstenberg's article Strict ergodicty and transformation of the torus and I'm stuck with the following construction. Define sequence $(v_k)_{k \in \mathbb{N}}$ as $v_1 =1, ...
2
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1answer
58 views

Bound for the sum of a finite sequence

Consider ${\bf c} = (a,b) \in \mathbb{R}^2$ with $0< \|{\bf c}\| < 1.$ Let $n \in \mathbb{N} $ and define \begin{align*} F_{n}(k) & := \frac{ [a + x_{n}(k)]^2}{ [a + x_{n}(k)]^2 + [b + ...
4
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3answers
136 views

To prove $\prod\limits_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$

Prove $$\prod_{n=1}^\infty\cos\frac{x}{2^n}=\frac{\sin x}{x},x\neq0$$ This equation may be famous, but I have no idea how to start. I suppose it is related to another eqution: (Euler)And how can I ...
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2answers
23 views

Using the denseness of trigonometric polynomials to prove the following

$f:[0,2\pi] \to \mathbb R$ is a continuous function. For every trigonometric function $T(x)=\sum_{k=0}^n a_k\cos(kx)+b_k\sin(bx)$, we have $\int_0^{2\pi}f(x)T(x)dx=0$. We need to prove $f=0$(and I ...
4
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2answers
131 views

Is there a way to simplify a sum of cosecants?

A problem I have been working on recently results in a sum of cosecant terms. Specifically, $f(n) = \sum_{k=1}^n \csc \frac{\pi k}{2n+1}$ $g(n) = \sum_{k=1}^n [(-1)^{k+1}(\csc \frac{\pi k}{2n+1})]$ ...
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2answers
49 views

Sum of the area of infinite similar equilateral triangles

How would I solve for the side depicted in the picture?
0
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1answer
24 views

Cauchy product when indices don't match

I want to find the first $4$ coefficients of the Maclaurin series of $\tan(z)$ by multiplying by $\cos(z)$ and using a Cauchy product. Letting $\tan(z)=\sum\limits_{k=0}^\infty c_kz^k$ and ...
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0answers
10 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$. [duplicate]

I have a sum of a series of trig functions as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. Here, we can assume $t$ is a small number and $t \neq 0$. what is the upper ...
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0answers
24 views

Lip $\alpha$ trigonometric series

Assume we have a trigonometric series $$ f(x)=\sum_{n=1}^{\infty} a_n\sin nx \in \text{Lip }\alpha, \, 0<\alpha <1. $$ Is there anything we can say about the series $$ g(x)=\sum_{n=1}^{\infty} ...
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2answers
41 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$?

I have a sum of a series of trig function as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. I am looking for the upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j ...
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0answers
62 views

Is $\sum_{k=2}^{\infty}{\sin(k!)}/{(k\log(k))}$ convergent or divergent? How can I prove it?

How can I prove that $~\displaystyle\sum_{k=2}^{\infty}\frac{\sin(k!)}{k\log(k)}~$ converges ? Both Leibniz's criterion and Dirichlet's test seem rather inadequate for handling this particular task. ...
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0answers
29 views

Trigonometry - word problem - functions cosθ = adjacent/hypotenuse

I'm not sure what trigonometric equation I should use for this problem: At a certain instant, a ship was 5km south of a lighthouse. The ship was travelling westward and after 30 minutes it's bearing ...
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3answers
56 views

Can we subtract a trigonometric term from a polynomial?

Can we find the root of a function like $f(x) = x^2-\cos(x)$ using accurate algebra or do we need to resort to numerical methods approximations? thanks.
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0answers
15 views

Approximation of functions by trigonometric polynomials?

I'm seemingly not understanding the fact that a continuous function $f$ that is periodic on [0, 1) can be approximated by a trigonometric polynomial of the form $$g(x) = \sum_{n = 0}^k c_n e^{2\pi ...
1
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1answer
31 views

Trigonometry system - complex conjugate

I have the following function $$e_n(t) = e^{2\pi int}, t \in R, n \in Z$$ Could anyone explain how one can go from this: $$e_m(t) \bar e_n(t) $$ to $$e^{-2\pi i(m - n)t}$$ Shouldn't it be $e^{2\pi ...
0
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2answers
32 views

Inverse Sine and cosine

$\arcsin(\cos(x))=1/2$ Find $x$. I got $-1/2$ or $2\pi-1/2$, but I don't know the correct answer. I tried graphing unit circle.
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2answers
80 views

Evaluating the sum $ \sum_{n = 1}^{44} {\sin^{2}}(n^{\circ}) ~ {\cos^{2}}(n^{\circ}) $.

I want to find the sum $$ {\sin^{2}}(1^{\circ}) ~ {\cos^{2}}(1^{\circ}) + {\sin^{2}}(2^{\circ}) ~ {\cos^{2}}(2^{\circ}) + {\sin^{2}}(3^{\circ}) ~ {\cos^{2}}(3^{\circ}) + \cdots + ...
1
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1answer
64 views

Summation of $\tan^{-1}$ series

I am given $$S=\sum\limits_{n=1}^{23}\cot^{-1}\left(1+ \sum\limits_{k=1}^n 2k\right)$$ On expanding the sigma series becomes $$S= 23\cot^{-1}(3)+22\cot^{-1}(5) + \cdots + \cot^{-1}(47)$$ And in tan ...
4
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1answer
45 views

Why does a fourier series have a 1/2 in front of the a_0 coefficient

I am reading up on the fourier series, and I keep seeing it as being defined as: $$ f(\theta)= \frac{1}{2}a_0 + \sum_{n=1}^{\infty}(a_n \cos(n\theta) + b_n \sin(n\theta)) $$ where $$ a_n = ...
2
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1answer
183 views

How to find the sum?

How to find $$\cot^2\frac{\pi}{2m+1}+\cot^2\frac{2\pi}{2m+1}+\cot^2\frac{3\pi}{2m+1}+\ldots+\cot^2\frac{m\pi}{2m+1}?$$ Of course, the number $m$ is assumed to be a positive integer.
0
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1answer
44 views

General form of $\sum_{i=1}^{k} \frac{1}{k}\tan(\frac{i\theta}{k})$

I have tried calculating this equation by setting a large value of $k$. given $\theta$, $$S = \sum_{i=1}^{k} \frac{1}{k}\tan\bigg(\frac{i\theta}{k}\bigg)$$ When increasing $k$, $S$ seems converging ...
1
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1answer
23 views

Trigonometical sum from Fourier analysis

(Edit) Note: $a\in \mathbb{R},0<a\leq\pi$. Also, the sum skips $n=0$ (that's where the other term comes from). Working through a Fourier analysis exercise I've got stuck in a clearly ...
4
votes
5answers
103 views

Evaluating $(1+\cos\frac{\pi}{8})(1+\cos\frac{3\pi}{8}) (1+\cos\frac{5\pi}{8})(1+\cos\frac{7\pi}{8})$

How to find the value of $$\left(1+\cos\frac{\pi}{8}\right)\left(1+\cos\frac{3\pi}{8}\right) \left(1+\cos\frac{5\pi}{8}\right)\left(1+\cos\frac{7\pi}{8}\right)$$ I used: $ 1+ \cos ...
0
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3answers
93 views

Deducing a $\cos (kx)$ summation from the $e^{ikx}$ summation [duplicate]

I'm trying to solve So far I've done the first part, evaluating the summation ; where a is just n. I'm not sure where to go from here or what it even means deduce the second summation. I ...
2
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2answers
48 views

Integral $\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$

Is this integral known to have a closed form? $$\int_0^{\infty}\cos(a_0+a_1x+a_2x^2)\frac{1}{x^2+\frac{1}{4}}dx$$ Is there anything special about it?
4
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2answers
157 views

Determine the limit of a series, involving trigonometric functions: $\sum \frac{\sin(nx)}{n^3}$ and $\frac{\cos(nx)}{n^2}$

I have $$\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}.$$ I did prove convergence: $0<\theta<1$ $$\left|\frac{\sin((n+1)x)n^3}{(n+1)^3\sin(nx)}\right|< \left|\frac{n^3}{(n+1)^3}\right|<\theta$$ ...
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0answers
29 views

Fourier series question - represent $x$ as a series of $\cos$

I was asked to represent $f(x)=x$ in $(0,\pi)$ as a sum of $\cos$ functions, using fourier series. I couldn't solve it on my own, but here is what the teacher did, and I don't fully understand why ...
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4answers
159 views

Number of iterations to reach cosine's fixed point

I was messing around with my calculator the other day when I saw something interesting happen. Whenever I repetitively took the cosine of any number, it always ended up on a particular number ...
6
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1answer
130 views

Is there any identity for $\sum_{k=0}^{n-1}\tan(x+ka) $??

I found this series $$ \sum_{k=0}^{n-1}\tan\left(\theta+\frac{k\pi}{n}\right)=−n\cot\left(\frac{n\pi}{2}+n\theta\right) $$ but it's not what I need.
1
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1answer
37 views

A trigonometric series with differences of cosines of cube roots

To be honest I don't know how to start working with this series. I can't even tell if the limit of the general term is $0$. Any help would be appreciated. ...
6
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1answer
87 views

What is $k_{\text{max}}$?

If $[1-\cos x][1 - \cos 2x][1 - \cos 3x] = k\ ; 0º < x < 90º$ Find $k_{\text{max}}$ I have no idea how to solve this I've got $8\left[\sin\left(\frac{x}{2}\right)\times\sin x ...
6
votes
4answers
173 views

How to compute $\sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)$

I find this problem on facebook group. $$\mbox{Is it possible to find exact value of}\quad \sum_{n\ =\ 1}^{\infty}\arctan\left(\,\frac{3n^{2}}{ 2n^{4} - 1}\,\right)\ {\large ?}. $$ I think this is ...
0
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1answer
35 views

Can you raise trigonometric functions to a non-integer power?

I don't inmediately see any reason why you could not yet I have never come across it. For any answer given reasoning would also be appreciated! Thank you
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3answers
38 views

Computing the trigonometric sum $ \sum_{j=1}^{n} \cos(j) $

I have a task to compute such a sum: $$ \sum_{j=1}^{n} \cos(j) $$ Of course I know that the answer is $$ \frac{1}{2} (\cos(n)+\cot(\frac{1}{2}) \sin(n)-1) = \frac{\cos(n)}{2}+\frac{1}{2} ...
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3answers
83 views

Prove using De Moivre's formula,that $\sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)}$

I've been asked to prove that: $$ \sum\limits_{k=0}^{n}\sin(kx)=\frac{1}{2}\cot(x/2)-\frac{\cos(nx+(x/2))}{2\sin(x/2)} $$ When $0<x<2\pi$. I know there are many similar posts on this site, but ...
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0answers
10 views

Product of truncated Fourier series

Given the truncated power series (trigonometric polynomials) $$ a(t)=\sum_{k=-P}^P A_k e^{ik\omega t} ,\quad b(t)=\sum_{k=-Q}^Q B_k e^{ik\omega t} $$ I am looking for formulas for the product and ...
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4answers
46 views

What is the approximation of trigonometric function by simple function

for $f(x)=\sin x$, $g(x)=\cos x$, $h(x)=\tan x$, What is the approximation of each function by using simple function?
1
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1answer
44 views

The asymptotic behavior of $\sum_{n=1}^\infty\frac{1-\cos(x4^n)}{2^n}$ as $x\to 0$

Is there a way to show that for small $x$'s $$\sum_{n=1}^\infty\frac{1-\cos(x4^n)}{2^n}\le c\sqrt x$$ I tried Taylor expansion of $\cos$ and square root... Thank's
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0answers
41 views

Evaluating $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$ without Complex Numbers [duplicate]

Alright, so the standard way to evaluate $\sum\limits_{k=0}^n\cos(kx)$ and $\sum\limits_{k=0}^n\sin(kx)$, is to respectively take the real and imaginary part of $$\sum_{k=0}^n{\rm e}^{ikx}={\frac ...
3
votes
4answers
293 views

Does $\sum_{x=1}^\infty\sin(x)$ converge?

I received a task to find out whether the following series converges: $$\sum_{x=1}^\infty\sin(x)$$ On first look it seems simple, but as I keep thinking about it, there's not a single lemma or ...
1
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2answers
21 views

Reverse of an iterative function

Here, i have a function for an iterative series. Next value = x + sin(x). converging on a value I want to make it so that i can find the current value, when i know the convergence value, The only ...
2
votes
2answers
51 views

What would be a power series for $f(z)=\sin(z)$ centered at $1$?

Everything is in the question! I've seen loads examples like "centered at $\pi$, $\pi/2$,... But $1$ would make everything much different... I've tried to work this way: $\sin(z) = \sin((z-1)+1) = ...
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2answers
61 views

How do I evaluate $\cos(x) + \cos (2x) +\cos (3x) + … + \cos (nx)$? [duplicate]

How to evaluate the above expression and express the answer in terms of $n$ and $x$?
0
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2answers
21 views

Find the values of $a$ and $b$ ~ Trigonometry

The function $f$, where $f(x) = a \sin x+b$, is defined for the domain $0 \leq x \leq 2\pi$. Given that $f(\frac{1}{2}\pi)=2$ and that $f(\frac{3}{2}\pi)=-8$, find the values of $a$ and $b$. I know ...
0
votes
1answer
41 views

Sum of trigonometric series $\sum_{m=1}^{N-1} \frac{\sin(4\pi mk/N)}{\sin ^2 (\pi m/N) }$

Anybody has some ideas to prove the following identity? \begin{equation} \sum_{m=1}^{N-1} \frac{\sin(4\pi mk/N)}{\sin ^2 (\pi m/N) }= 0 \end{equation} where $N$ is an integer greater than $1$, $k$ ...
1
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2answers
34 views

What function does this trigonometric series represent?

I wonder what known function does this trigonometric series represent? $$ f(x)=\sum_{k=1}^\infty \frac{1}{k^{3/2}}\sin{(kx)} $$ with $$ -\pi\le x\le \pi $$ and $$ f(x)=f(x+2\pi) $$ And how is it ...
1
vote
2answers
27 views

How to determine if this converges?

I find this one to be hard, I tried expanding $\cos$ into a Taylor series, but that didn't work out well because I couldn't apply $p$-series... $$ \sum^{\infty}_{n=1} n (1-\cos(\pi/n)) $$
1
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3answers
57 views

How to determine if this series converges?

Does this series converge? I tried using limit comparison, and I don't know what to try next... $$\sum^{\infty}_{n=1}(1-\cos (\pi/n)) $$
2
votes
1answer
63 views

Nowhere differentiability of Weierstrass function

It's again from Tao's book. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic interval ...
1
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0answers
47 views

Harnack's curve theorem for zero sets of real trig polynomials in two variables?

Is there a result like Harnack's curve theorem for real trig polynomials which gives bounds for the number of connected components the zero set can have in terms of the degree? More specifically, let ...