For questions about or related to trigonometric series.

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4answers
35 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
42 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
23 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
229 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 ...
19
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1answer
367 views

Proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$

Show that $$\sum_{n=1}^{\infty}\frac{\sinh\big(\pi n\sqrt2\big)-\sin\big(\pi n\sqrt2\big)}{n^3\Big({\cosh\big(\pi n\sqrt2}\big)-\cos\big(\pi n\sqrt2\big)\Big)}=\frac{\pi^3}{18\sqrt2}$$ I have no ...
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0answers
39 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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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 ...
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0answers
97 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 ...
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2answers
480 views

How do I find the direction angle of a vector?

Let $v= -3 i-10j$. I found the dot product to be $109$ and the magnitude to be $\sqrt{11881}$. I divided it out and it came out to be one. I don't know what I'm doing wrong. Please help.
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2answers
49 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
44 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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2answers
53 views
0
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1answer
30 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
16 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 ...
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1answer
49 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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2answers
28 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
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2answers
25 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
54 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... $$\sum^{\infty}_{n=1}(1-\cos (\pi/n)) $$
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0answers
30 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 ...
3
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0answers
33 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
62 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
16 views

Use of small approximations to get the following formula

I cannot see how the following formula has been found $\displaystyle \cos(\theta + \frac{v\epsilon^2 \Omega}{L}+O(\epsilon^4))-\cos(\theta) = -\epsilon^2\frac{\Omega v}{L}\sin(\theta)+O(\epsilon^4)$ ...
6
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4answers
497 views

A sine integral

The integral \begin{align} \int_{0}^{\pi/2} \frac{ \sin(n\theta) }{ \sin(\theta) } \ d\theta \end{align} is claimed to not have a closed form expression. In this view find the series solution of the ...
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2answers
57 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 $\alpha \ne 1$. ...
0
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1answer
26 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 ...
0
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1answer
30 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
14 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
277 views

Extensions of Ramanujan's Cos/Cosh Identity

The Ramanujan Cos/Cosh Identity is stated here as $$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+ \left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}= ...
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1answer
23 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 ...
3
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2answers
486 views

Does the sum $\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}$ converge for $\alpha > \frac{1}{2}$ and $x \in [0,2 \pi]$?

Does the series $$ \sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}, $$ converge for all $\alpha > \frac{1}{2}$ and for all $x \in [0,2 \pi]$? It is obvious that it does when $\alpha > 1$, ...
6
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0answers
85 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
133 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: $\sum_1^\infty{a_k}$ ...
3
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2answers
232 views

Does the series $\sum_{n\ge1}\left(-1\right)^n\frac{\cos\left(\alpha n\right)} {\sqrt n}$ converge?

Does the series $\sum_{n\ge1}\left(-1\right)^n\frac{\cos\left(\alpha n\right)} {\sqrt n}$ converge ? I tried to deal with this problem this way. Let $S_k$ be a sequence of partial sums of the given ...
5
votes
9answers
441 views

Prove $ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $

$$ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $$ Wolframalpha shows that it is a correct identity, ...
3
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1answer
94 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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0answers
34 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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2answers
47 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}= ...
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1answer
64 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
42 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
39 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
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1answer
49 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 ...
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1answer
66 views

Hard Trigonometric Equation

its possible to solve a equation like $$\prod^{45}_{k = 0} \left( 1 + \tan \frac{k \pi }{180} \right) = \left[ \log_{\frac{\sqrt{6}}{3}} \big| \sin(2x)\big| \right]^{\frac{9}{2}\sin(2x) + 20}$$ ...
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2answers
125 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 ...
0
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1answer
54 views

Solve for x: sin2 x − cos2 x = sin x, −π ≤ x ≤π

I have to solve for x using the domain of −π ≤ x ≤π sin^2 x − cos^2 x = sin x I tried changing cos^2 x to 1 - sin^2 x so it would be sin^2 x - 1 + sin^2 x = sin x making it, 2sin^2 x - 1 = sin x ...
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0answers
37 views

Using Algebra with Trig Functions

Using Algebra with Trig Functions I'm trying to find the correct 1 second audio signal I would need to apply to a 1 second known noise signal to have the output signal be a sin wave. The basic ...
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3answers
75 views

Evaluation of a trigonometric series

$$ \mbox{Question: Evaluate}\quad \tan^{2}\left(\pi \over 16\right) + \tan^{2}\left(2\pi \over 16\right) + \tan^{2}\left(3\pi \over 16\right) + \cdots + \tan^{2}\left(7\pi \over 16\right) $$ What I ...
0
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0answers
50 views

Trigonometry - proving a rule

I was given that $\sin a + \sin b + \sin c \cdots$ is equal to: (where $a,b,c$ are in arithmetic progression) $$\frac{\sin\frac{a + c}{2}\sin\frac{nb}{2}}{\sin{b/2}}$$ Here $a$ is the first term of ...
0
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0answers
27 views

Help in finding a book

That is my problem: i have this series: $\sum_{k=0}^{\infty} \psi_{k}(h)\cos(k\theta)$. I need to see to what it converges assuming something on the functions $\psi_{k}(h)$. There is a book or a site ...
0
votes
2answers
58 views

Proving that $\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$.

Prove: $$\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$$ Thanks a lot!! I tried: With Euler's identity I can get $\sin x= \dfrac{e^{ix} - e^{-ix}}{2i}$ and the ...
4
votes
2answers
182 views

Find the limit of $\prod_{k = 4}^{\infty}\cos\left(\pi \over k\right)$

Find the limit of $$\prod_{k = 4}^{\infty}\cos\left(\pi \over k\right)$$ The limit does exist, but I can not get it. Thanks Willie-Wong & Lee Mosher for correcting the expression.