Questions tagged [trigonometric-series]

For questions about or related to trigonometric series, i.e. series of the form $a_0 + \sum_{n = 1}^{\infty} (a_n \cos{nx} + b_n \sin{nx})$ or $\sum_n c_n e^{inx}$.

Filter by
Sorted by
Tagged with
0
votes
0answers
47 views

Double angle formula broken? $\cos{(\frac{n\pi}{2})} = 0, 1, 0, 1, …, 0, 1, … \neq \cos{(\frac{n\pi}{2})} = 0, -1, 0, 1, 0, -1 … 0, -1, 0, 1…$

First post here, please forgive me if my formalism is not the best. I noticed something while working on some homework trying to simplify an equation and I am quite confused by it: $\cos{(2\theta)} = ...
0
votes
1answer
43 views

Is it possible to determine for what portion of the domain, a function $f_1(x)\cdot f_2(x)\cdot f_3(x)$ is positive?

We have a one-variable function for $x>0$ where $f_i(x)$ are some periodic trigonometric functions $F(x)=f_1(x)\cdot f_2(x)\cdot f_3(x)$ We know that $f_1(x)$ is positive for $\frac23$ of its ...
0
votes
1answer
50 views

Can't find exact value of $x = \frac{\pi}{\cot^{-1}(\frac{4\pi}{x})}$

I can't find the exact value of x for the equation $x = \frac{\pi}{\cot^{-1}(\frac{4\pi}{x})}$. I tried typing it into wolfram alpha and it only gave me a decimal approximation, approximately equal to ...
0
votes
3answers
43 views

Expand the function $[\cos(x^3)]^\frac{-1}{2}$ into Taylor series around $x=0$ up to $O(x^6)$

I need help in expansion of the following function into Taylor Series up to $O(x^6)$ $$[\cos(x^3)]^\frac{-1}{2}$$ The things I've already tried are expansion of $u^\frac{-1}{2}$ by assuming $u=cos(x^3)...
-1
votes
0answers
20 views

Explicit expression for a recursive sequence containing a trigonometric function

I would like to determine the explicit function of a recursive sequence. The recursive sequence is: $$x_{n+1} = \tanh(x_n)$$ I am not sure how to approach this problem. In another post I saw that ...
0
votes
0answers
29 views

Partial fraction expansion of $\sec x$

I have been searching for the proof of this and according to wolframe alpha $\sec(x)=4\pi \sum\limits_{n=0}^{\infty} (-1)^n \frac{(2n+1)}{(2n+1)^2 \pi^2-4x^2}$, for all $-\frac{1}{2}+\frac{x}{\pi}\...
1
vote
0answers
27 views

How to estimate this trigonometric sum?

I am reading Gregory F. Lawler's Random Walk and the Heat Equation. In page 39-40 the author considers a set the following problem: Let $N$ be a positive integer, $N\geq 2$. Let $A_N=\{(x_1,x_2): x_i=...
2
votes
0answers
46 views

If $x\cot x=a_0+a_2x^2+a_4x^4+\cdots$, then $\frac{a_{2n}}{1!}-\frac{a_{2n-1}}{3!}+\cdots+\frac{(-1)^na_0}{(2n+1)!}=\frac{(-1)^n}{(2n)!}$

If $x\cot x=a_0+a_2x^2+a_4x^4+\cdots$, then show that $$ \dfrac{a_{2n}}{1!}-\dfrac{a_{2n-2}}{3!}+\dfrac{a_{2n-4}}{5!}-\cdots+\dfrac{(-1)^na_{0}}{(2n+1)!}=\dfrac{(-1)^n}{(2n)!} $$ This problem is from ...
3
votes
1answer
108 views

Inequality regarding L^6 norm of a trigonometric polynomial

I am reading Bourgain's paper "Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations", and am stuck on the following ...
2
votes
1answer
73 views

Need help integrating a trigonometric function over an interval

I found an equation for calculating the perimeter of an ellipse and need help converting it to its series. $$ \operatorname{p}\left(a,b\right) = 4a\int_{0}^{\pi/2}\sqrt{\,{1 − \left(\frac{\sqrt{a^{2} -...
2
votes
1answer
57 views

Alternating series of powers of Cosine-squared

I am looking for a nice(er) proof of the following identity: $$\sum_{k=1}^{n-1} (-1)^k \cos^{2j}\frac{\pi k}{2n} = -\frac{1}{2}, ~ 0<j<n,~j,n\in \mathbb{Z}^+.$$ I have been able to prove it ...
1
vote
1answer
18 views

Odd Periodic Extension to Obtain Fourier Series

I'm asked to obtain the Fourier series for $f(x)=x|x|$ on the interval $[-1,1]$. I'm thinking it's easier to do an odd periodic extension of $x^2$ on $[0,1]$. My book says this is given by $$f(x)=\...
0
votes
0answers
46 views

Theorem 8.14 in Rudin's “Principles of Mathematical Analysis”

I am reading Rudin's "Principles of Mathematical Analysis" and I am stuck at the following theorem Theorem 8.14: If, for some $x$, there are constants $\delta > 0$ and $M < \infty$ ...
0
votes
1answer
34 views

Divisibility of Chebyshev Polynomials

I was trying to solve a problem involving an Insect crawling on the Cartesian/Coordinate Plane. We have an insect on the origin of the coordinate plane, who remembers a particular angle $\theta.$ We ...
1
vote
1answer
55 views

Solving a nonlinear first order ODE

In this text https://www.springer.com/gp/book/9781461454762 on p. 95, it has the following The text does not explain how to get to this solution. $B, C, D$ are arbitrary constants. With $A>0$, I ...
0
votes
2answers
59 views

Proof explanation for a sum of sines [duplicate]

Looking through proofs and there's one part I'm confused about, that if solved will get me straight to the answer. I need to show that $\sum_{m=0}^{N-1} \sin((m+\frac{1}{2})x) = \frac{\sin(Nx/2)^2}{\...
2
votes
1answer
17 views

Why does the cumulative summation of cos(n^2) have a strange symmetry?

I was messing around in Desmos and typed this in: $\displaystyle{\sum_{n=1}^{floor(x)}\cos(n^2)}$ It produces this graph in the $XY$-plane (keep in mind the axis ratios): $XY$-plane" /> It appears to ...
1
vote
1answer
26 views

Need clarification on the complex form of Fourier series

I wish to ask you guys to fill in a few steps for the derivation of complex form of Fourier series. This is taken from "Fourier series" of Tolstov (Dover publication). $$f(x)\sim c_0+\sum_{n=...
1
vote
1answer
86 views

“Summing” the series $\sin(x)+\frac{1}{2}\sin(2x)+\frac{1}{3}\sin(3x)+\frac{1}{4}\sin(4x)+…$ [duplicate]

"Summing" the series $\sin(x)+\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)+\dfrac{1}{4}\sin(4x)+...$ Pose $$S=\sin(x)+\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)+\dfrac{1}{4}\sin(4x)+...$$ $$C=\...
3
votes
2answers
78 views

“Summing” the series $\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-…$

"Summing" the series $\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-\dfrac{1}{4}\sin(4x)+...$ Pose $$S=\sin(x)-\dfrac{1}{2}\sin(2x)+\dfrac{1}{3}\sin(3x)-\dfrac{1}{4}\sin(4x)+...$$ $$C=\...
0
votes
2answers
52 views

Show that the series $\sum_{n=0}^{\infty}\dfrac{\cos(nx)}{\cos^{n}(x)}$ is divergent

In this thread, I use the "C+iS" summation method for trigonometric series to sum it: lab bhattacharjee pointed out that: $\left|\dfrac{e^{ix}}{\cos(x)}\right|=|\sec(x)|\geq1$ What I don't ...
1
vote
2answers
68 views

Help to sum these two series $\sin(x)-\sin(2x)+\sin(3x)…$

I am reading Euler's paper entitled "Subsidium Calculi Sinuum" and he wrote down some "sums" for these trigonometric series: \begin{align}S &= \sin(x)-\sin(2x)+\sin(3x)-\sin(4x)...
1
vote
2answers
68 views

Summing the trigonometric series $\sin(a)+\frac{1}{2}\sin(2a)+\frac{1}{2^2}\sin(3a)+\frac{1}{2^3}\sin(4a)…$

This is a problem of Loney's "Plane trigonometry, part 2" $$\sin(a)+\frac{1}{2}\sin(2a)+\frac{1}{2^2}\sin(3a)+\frac{1}{2^3}\sin(4a)...=\sum_{n=0}^{+\infty}\frac{1}{2^n}\sin[(n+1)a]$$ Here is ...
1
vote
1answer
73 views

Is it possible to prove that $\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^a}\geq 0$ for $a\geq 1$ and $x\in [0,\pi]$?

I was trying to prove that $$\sum_{k=1}^{\infty}\frac{\sin(kx)}{k^a}\geq 0,$$ for $0\leq x\leq \pi$, and $a\geq1$. For $a$ being an odd integer, this is not really a problem, as the sums may then be ...
2
votes
1answer
115 views

A trigonometric function comparing with $0$

Short Version We need to solve this problem: Prove that function $$f(x) = 2\sin\left(\frac{\pi}{x+\frac{1}{x}+1}\right)-\sin\left(\frac{\pi}{x^{2}+x+1}\right)-\sin\left(\frac{\pi}{\frac{1}{x^{2}}+\...
1
vote
0answers
41 views

Series for $f(x)=a\ln(1+b\sin(cx+df(x)))$?

This question is inspired by the two questions here and here. The answers to these questions show several ways to obtain approximations respectively give explicit fourier expansions for the functions ...
2
votes
0answers
20 views

Using Bessel's DE to prove conditions

Let $y_p(x)$ be a nontrivial solution of Bessel's DE of order $p$. Let $x_1$ and $x_2$ be successive zeroes. By using the normal form of the DE and comparing this to the well-known solutions of $$y''+...
2
votes
1answer
58 views

Need help evaluating this infinite sum involving a sine function and squared denominator

I need major help in evaluating the following infinite sum: $$\sum_{k=1}^{\infty} \frac{k\sin(kx)}{(k^2+a^2)^2}\tag{1}\label{whatiwant}$$ where a is a constant. I know that (from Gradshteĭn et al, ...
2
votes
3answers
99 views

How to find the minimum value of cos(2x)+cos(4x)+cos(6x)+cos(8x)+…+cos(20x)

I want to find the minimum value of the series cos(2x)+cos(4x)+cos(6x)+cos(8x)+...+cos(2nx). x could be 2pit. Anyone can share a method of how to determine the minimum value of the series?
1
vote
1answer
24 views

Trignometric functions and their polynomial forms

The trigonometric values also has a infinite series which I had learnt from the topics of Limits Sin x = x - x³/ 3! + x⁵/ 5!....... There are formulas like these for other trigonometric functions ...
1
vote
3answers
77 views

A problem on inverse trigonometric function

How to sum the following series: $$S=\cot^{-1}2+\cot^{-1}8+\cot^{-1}18+\cot^{-1}32+\cdots+\cot^{-1}∞$$ My attempt better to say a solution was,\begin{align}S &=\sum_{n=1}^{∞}\cot^{-1}2n^2\\ &=\...
2
votes
2answers
114 views

Show that $e^{ix}=\cos(x)+i\sin(x)$ using the Fourier Series only

As stated in the title. Any arbitrary function can be expressed as $$f(x)=\frac{a_0}{2}+\sum^{\infty}_{n=1}(a_n\cos(nx)+b_n\sin(nx)) \tag{1}$$ where $$a_n=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)dx \...
0
votes
0answers
15 views

How to interpolate a list of points/complex numbers with a single trigonometric polynomial.

If I have a table of points or complex numbers, is there a way that I can get a single function to plot through all of them? WolframAlpha does this with its "popular curves" which seem to be ...
0
votes
4answers
70 views

Prove $\sin(\pi/2-x)\cot(x+\pi/2) = -\sin(x)$

I've managed to use cofunction identities to get the left side of the equation to (pi/2-x)-tanx. From here, I keep ending up at -cotx? I'm pretty sure that's wrong, and I have no idea how to to go ...
10
votes
6answers
420 views

Find the Maximum Trigonometric polynomial coefficient $A_{k}$

Let $n,k$ be given positive integers and $n\ge k$. Let $A_i, i=1, 2, \cdots, n$ be given real numbers. If for all real numbers $x$ we have $$A_{1}\cos{x}+A_{2}\cos{(2x)}+\cdots+A_{n}\cos{(nx)}\le 1$$ ...
0
votes
1answer
36 views

how to prove the multiplication formula for the cotangent function

I could not find the proof of the following formula in the internet: $$\frac{k\tan x}{\tan kx}=1+\frac{1}{2}\sum_{0<j<k} \frac{\tan x}{\tan (x+\frac{\pi j}{k})}+\frac{\tan x}{\tan (x-\frac{\pi j}...
1
vote
1answer
61 views

Evaluate:$\sum_{n=2}^{\infty}\frac{\tan \theta_{n}}{3^n\left(3-\tan^2\theta_{n}\right)}$

Evaluate:$$\sum_{n=2}^{\infty}\frac{\tan \theta_{n}}{3^n\left(3-\tan^2\theta_{n}\right)}$$ where $$\theta_{n}=\frac{\theta}{3^n}$$ and $0<\theta<\pi$ I did try to find relation between $\tan 3x$ ...
4
votes
2answers
82 views

$\frac{d}{dx}(\sin(x^{\frac{1}{3}}))$ from first principle

The question contains a hint: at the appropriate point use the result $a^{3} - b^{3} = \left(a - b\right)\left(a^{2} + b^{2} + ab\right)$. $$ \frac{{\rm d}}{{\rm d}x}\sin\left(x^{1/3}\,\right) $$ My ...
0
votes
1answer
35 views

Expansion of trigonometric function squared (with two different method)

A series expansion of $\cos(x)$ is $$ \cos(x)=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n} $$ I want to get the series expansion of $\cos^2(x)$, but using two different method I get different result ...
0
votes
1answer
42 views

Question about Fourier cosine series

I'm trying to find Fourier cosine series for function $$f(x) = \begin{cases} \frac{\pi}{2}-x,\quad &\text{if } x\in\left[0,\frac{\pi}{2}\right) \\ \pi,\quad &\text{if } x\in\...
1
vote
1answer
37 views

Is there a way to solve $\arcsin\left(\frac{d}{r}\cdot \sin(x)\right)=\phi-x+\frac{\theta}{2}$ for $x$?

While solving a geometry problem, I derived the following equation: $$\arcsin\left(\frac{d}{r}\cdot \sin(x)\right)=\phi-x+\frac{\theta}{2}$$ And, in order to solve such problem, I need to solve for $x$...
1
vote
2answers
32 views

Proving the Fourier equation: $\frac{1}{2} L - x = \frac{L}{\pi}\sum^\infty_{n=1}\frac{1}{n} \sin\frac{2n\pi x}{L}$

The following question is about Fourier series, specifically about extension of the Fourier Series to arbitrary intervals: "Show that: $$\frac{1}{2} L - x = \frac{L}{\pi}\sum^\infty_{n=1}\frac{1}{...
1
vote
0answers
32 views

What is the Jacobi-Anger expansion of the $k$'th functional iterate of the sine function?

The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \sum_{n=1}^{\infty} J_{2n-...
0
votes
1answer
52 views

Taylor expansion of $\lvert\sin(x)\rvert$

Is it possible to generate a Taylor expansion of $\lvert\sin(x)\rvert$? I understand that this is not possible around the point $0$, since $\frac{d}{dx}\lvert\sin(x)\rvert$ is undefined at $x = n\pi$. ...
0
votes
0answers
30 views

How do you find the intersection of these two trig curves?

How do you find the intersection of these two trig curves? $ \cos(x_1) + \cos(x_2) + \cos(x_3) = 1 \\ \sin(x_1) + \sin(x_2) + \sin(x_3) = 0 $ Alternatively, how would you find a general solution for $...
0
votes
0answers
22 views

Determine the convergence or divergence of the series

I'm having some trouble trying to get started on this problem. $\sum_{n=1}^{\infty} [\sin(1/(2n)) − \sin(1/(2n+1))]$. My initial thoughts were to try to work the series as two individual series, see ...
0
votes
2answers
39 views

Simplification of Complex number summation [closed]

How does the author of this book make these simplifications from left to right? It doesn’t seem obvious. Here $h,\alpha$ are real numbers and $N$ is a natural number $$ \left|\sum_{n=1}^{N}e^{2\pi i h ...
3
votes
0answers
156 views

Does this erratically-behaved infinite sum converge?

I have been trying to find the convergence (and value) of this infinite sum: $$\sum_{n=1}^\infty \frac{\sin(n)^n}{\cos(en)}$$ The partial sums behave relatively unpredictably and at some point become ...
1
vote
0answers
63 views

Trigonometric polynomial is non-negative for all integer values of m

I'm working on a problem in acoustic scattering, where I have to calculate the torque exerted on a cylinder. The torque, $\tau(\phi,m)$, is a function of the "maximum phase" denoted by $\phi$...
0
votes
0answers
32 views

Somewhat open-ended question about graphs of certain functions involving sines and cosines

\begin{align} h(x) & = \begin{cases} +1 & \text{for } x\in (0,+\pi) \bmod 2\pi, \\ -1 & \text{for } x \in (-\pi,0) \bmod 2\pi. \end{cases} \\[8pt] g(x) & = \pm\arccos\left( \frac{\sin ...

1
2 3 4 5
16