For questions about or related to trigonometric series.

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1answer
25 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?
2
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4answers
634 views

Euler's formula, is this true?

Let me have a function such as $ f(k) = \exp( j 2 \pi k ) $, where $k$ is real value. $$ f(k) = \exp( j 2 \pi k ) = (e^{j 2 \pi} )^{k} =1^{k} = 1.$$ I think if this is true, there is no need to ...
2
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0answers
34 views

Problem with constants of $\sin(x)$ and using partial sums (Basel Problem)

I'm working on the Basel problem, and in my working I have $\sin(x)=Ax(x-\pi)(x+\pi)(x-2\pi)(x+2\pi)...$ But I think I have a problem with the coefficient of x - wouldn't it be infinitesimal, if we ...
3
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2answers
118 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
27 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
134 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 ...
5
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1answer
125 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.
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1answer
34 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
85 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
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4answers
166 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
33 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
35 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} ...
2
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3answers
68 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 ...
1
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4answers
44 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?
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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
3
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0answers
28 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
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4answers
277 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 ...
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2answers
16 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
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2answers
50 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
55 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 $x$?
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2answers
20 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
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1answer
35 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 $1$, $k$ ...
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2answers
32 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 ...
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2answers
26 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... $$ \sum^{\infty}_{n=1} n (1-\cos(\pi/n)) $$
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3answers
56 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
56 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 ...
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0answers
38 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 ...
4
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2answers
49 views

Solve equation for $0^\circ < x < 360^\circ$

Solve the following equation for $0^\circ < x < 360^\circ$ $$\cos(2x - 15^\circ) = -0.145$$ By finding out the cos inverse, I get $81.7^\circ$. Because $-0.145$ is negative, it lies on the ...
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0answers
100 views

Value of a Sine-Like Infinite Product

Does the following infinite product have a "nice" closed form? $$ P = \prod_{k=2}^{\infty} \left(\left(1 - \frac{1}{k^2}\right)^\dfrac{(-1)^k}{k}\right) $$ I know that without the power one could ...
3
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0answers
37 views

Trigonometric sum evaluation

Let $q$ a prime number and $1 \leq a<q$ a positive integer. We know from Ramanujan identity that $$\underset{h=1,\left(h,q\right)=1}{\overset{q}{\sum}}e^{2\pi ...
3
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1answer
63 views

Why do the averages of $\sin (p+\cos p )$ approach a positive limit?

$$S_1 = \sum_{k=1}^n (\sin \left[~ p(k) + \cos p(k)~ \right])$$ I wonder why this appears to give $$\frac{1}{n}S_2\sim 1/2 $$ Thanks for any insights or references. Edited in light of ...
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2answers
19 views

Use of small approximations to get the following formula

I cannot see how the following formula has been found $\displaystyle \cos(\theta + \frac{v\epsilon^2 \Omega}{L}+O(\epsilon^4))-\cos(\theta) = -\epsilon^2\frac{\Omega v}{L}\sin(\theta)+O(\epsilon^4)$ ...
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2answers
67 views

How can I find fifth root of unity?

I have no idea to do this question, how can I find the fifth root of unity? Question : Find all the distinct fifth root of unity. Let $\alpha$ be a fifth root of unity such that $\alpha \ne 1$. ...
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0answers
40 views

Estimation of trignometric polynomial and Lipschitz estimation

Let $\mathcal{T}_n$ denote the linear space of trigonometric polynomials of degree up to $n$ and $$E_n(f)=\inf_{P\in\mathcal{T}_n} ...
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1answer
32 views

MacLaurin of the Third-degree in sin(a*x)*cos(b*x) at given values

Alright so from my understanding MacLaurin is a special case of Taylor Series but at f(0). However my question involves solving the third degree of MacLaurin for $$f(x) = sin(a \times x)\times ...
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1answer
32 views

Is it possible to rearrange this for x?

Is it possible to rearrange $\tan(y) = \frac{\sin(x)}{\cos(x)+C}$ for x, where C is a constant? Thank you for any suggestions!
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0answers
16 views

Understanding this question in regards to Taylor Polynomials

I am looking at the following question, and I simply don't understand it. I've calculated the Taylor series for $f(x) = \arcsin(x)$ centered around $0$ of order $n=3$. Evaluating this series at $x = ...
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1answer
26 views

Cosine of the Natural Logarithm - Series Expansion

I am interested in a computable series expansion of the following equation: $f(n) = \cos(\log(n))$ Specifically, I am interested in real values of $n$ where $n>1$. From basic series definitions ...
6
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0answers
98 views

Solving a problem of Ramanujan's interest

I am Brian Diaz, and I am new to the math.stackexchange community. I have been struggling with attempting to find a closed form of the following series: $$ \varphi(\theta) = 1 + 2\sum_{n=1}^{\infty} ...
6
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2answers
137 views

How do I show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m))) + \cdots$ converges for all real numbers $m$?

I'm trying to show that the series: $\sin(m) + \sin(\sin(m)) + \sin(\sin(\sin(m)) + \cdots$ converges for all real numbers $m$. To be specific, the series is defined as follows: $\sum_1^\infty{a_k}$ ...
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0answers
43 views

How to calculate the sample autocorrelation coefficient of this time series?

Given : $\hat{\gamma}(h)=n^{-1}\Sigma_{t=1} ^{n-|h|}(x_{t+|h|}-\bar{x})(x_{t}-\bar{x})$ $\bar{x}=\frac{1}{n}\Sigma_{t=1} ^n x_{t}$ $\hat{\rho}(h)=\frac{\hat{\rho}(h)}{\hat{\rho}(0)}$ $x_{t}=c \cdot ...
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1answer
101 views

Measure of the set where a trigonometric polynomial with zero mean is non-negative

Suppose $f$ is a real trigonometric polynomial of degree $N$ with constant term $0$. What lower bounds can we place on the measure $\mu$ of the set $\{ t \in S^1 : f(t) \geq 0 \}$, independent of the ...
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2answers
51 views

Sum of Complex series

Let $\theta\in\mathbb{R}$ and $\theta \neq k\pi$ for $k\in\Bbb Z$. By summing a geometric progression show that $$1 + e^{2i\theta} + e^{4i\theta}+e^{6i\theta} + e^{8i\theta}= ...
1
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1answer
66 views

Sum of the trigonometric series

I'm studying de Moivre's theorem's application on the summation of trigonometric series. Here's what I have so far: \begin{align*} \sum_{k=0}^n \cos(k\theta)&= \text{Re}\sum_{k=0}^n e^{ki\theta} ...
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1answer
47 views

Trigonometric series problem: finding a second valid solution.

Given that I can do part of this question so here goes: Substituting $\theta=\frac{1\pi}{11}$ into LHS of given expression gives $$\cos\frac{1\pi}{11} + \cos\frac{2\pi}{11} + \cos\frac{3\pi}{11} ...
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2answers
54 views

Problem involving summing exponential series:

I can show the first part (i) (a), but the second part (b) i think it should be $S=\infty$ since the denominator is zero with that value of $\theta$. However, this is not the answer, any ideas? ...
2
votes
1answer
57 views

Sequence of trigonometric polynomials which converges to an integrable function

A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$. Can we find ...
1
vote
1answer
73 views

Hard Trigonometric Equation

its possible to solve a equation like $$\prod^{45}_{k = 0} \left( 1 + \tan \frac{k \pi }{180} \right) = \left[ \log_{\frac{\sqrt{6}}{3}} \big| \sin(2x)\big| \right]^{\frac{9}{2}\sin(2x) + 20}$$ ...
2
votes
2answers
129 views

Differentiability of the sum of the series $\sum_k \sin(kx)/k^2$

How to show the following: If $ f(x) = \displaystyle\sum_{k=1}^{\infty} \dfrac {\sin(kx)}{k^2} $, then show that $f(x)$ is differentiable on $(0,1)$ I guess it should be related to uniform ...