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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 $\sum_n c_n e^{inx}$.

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Minimize the max value from a sum of sines

I'm considering functions of the form $$ \operatorname{f}\left(t\right) = \sin\left(\omega_1 t - \phi_1\right) + \sin\left(\omega_2 t - \phi_2\right) + \cdots + \sin\left(\omega_n t - \phi_n\right) $$ ...
Arthur Prudius's user avatar
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0 answers
34 views

The conjugate of a cosine series is a sine series

In Katznelson's An Introduction to Harmonic Analysis the author defines the conjugate $\widetilde{S}$ of a trigonometric series $$S \sim \sum_{n = \infty}^\infty a_n e^{inx}$$ by $$\widetilde{S} \sim \...
approximate-identity's user avatar
0 votes
1 answer
75 views

An infinite nested radical [closed]

Can anyone help me in finding a closed form of the infinite nested radical here $$\left({\sqrt {4+\sqrt {4+\sqrt {4-\sqrt {4+\sqrt {4+\sqrt {4- ......\infty}}}}}}}\right)$$ The signs are as "+,+,-...
Rieman Tieman's user avatar
1 vote
2 answers
108 views

Proof for tan($A_1+A_2+A_3+\cdots+A_n)$ [duplicate]

Please check the example for clearer understanding of $S_k$ We all know the formula of tan($A_1+A_2+A_3+\cdots+A_n)=$ summation of tan of terms ($A_1,A_2...$) taken once (Let this be $S_1$) - ...
Rishwanth S V's user avatar
0 votes
1 answer
30 views

Alternate sum of Chebyshev polynomials

The problem is For all integer $n\ge1$, \begin{align}\frac{(-1)^n}{2^{n-1}}\left(\frac12+\sum _{k=1}^n (-1)^{k} T_k(x)\right)&=\prod _{j=0}^{n-1} \left(x-\cos \left(\frac{\pi  (2 j+1)}{2 n+1}\...
hbghlyj's user avatar
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0 answers
38 views

Simplify $\arctan(x + b)$ where $b$ is a constant positive integer?

I was wondering if it was possible to simplify $\arctan(x + b)$ so that I can factor out the $x$ from the expression entirely. I tried searching a rule for the $\arctan$ of a sum but came up empty-...
Arjun Krishnan's user avatar
2 votes
1 answer
95 views

Power sum of equi-spaced cosines

My question is to prove the following identity holds for all $n,k\in\mathbb N$ and $\tau\in\mathbb C$. $$\tag1\label1 \sum _{j=1}^{n} \cos ^k\left(\frac{2 \pi  j+\tau }{n}\right)=\frac{n}{2^k}\sum_{\...
hbghlyj's user avatar
  • 2,770
2 votes
3 answers
112 views

Check convergence of a trigonometric series

I have a problem with the following series: $$ \sum_{n=1}^{\infty}\sin \frac{1}{\sqrt{n}} \tan \frac{1}{\sqrt{n}}$$ My idea is to check if it's absolutely convergent. As $x\to 0$ we can use following ...
Cas's user avatar
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7 votes
3 answers
207 views

Behavior of function $\sum_{j = n}^\infty \frac{\sin^2((2j-1) \pi x)}{(2j-1)^2}$

For a positive integer $n$, define the function $$ F_n(x) = n^2 \sum_{j = n}^\infty \frac{\sin^2((2j-1) \pi x)}{(2j-1)^2}. $$ I am trying to understand the behavior of $F_n(x)$ in the following sense. ...
Drew Brady's user avatar
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0 votes
1 answer
69 views

Behavior of an Infinite Series

I've been studying infinite series recently and believe I came across a counterintuitive (at least to me) result in the past from a textbook that I can't seem to find now. Is it possible to show $$\...
Clayton's user avatar
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1 vote
1 answer
84 views

Is it true that $\min\{x, y\} = 2 \sum_{n \geq 1, n~\text{odd}} \frac{\sin(n x) \sin(ny)}{n^2}$?

Let $\lambda_j = \tfrac{4}{(2j - 1)^2 \pi^2}$ and $\psi_j(x) = \sqrt{2} \sin(\tfrac{2j-1}{2} \pi x)$ denote the eigenfunctions associated to the first-order Sobolev space. It is claimed in many texts ...
Drew Brady's user avatar
  • 3,673
1 vote
1 answer
48 views

prove that $\sum_{n = 1}^{\infty} \frac{|\sin(nx)|}{n}$ is divergent [duplicate]

I tried much to show that the following series is divergent for $x \in (0,\pi)$ $\sum_{n = 1}^{\infty} \frac{|\sin(nx)|}{n}$ My opinion was to use the comparison test with series $\frac{1}{n}$ but I ...
schneiderlog's user avatar
1 vote
0 answers
73 views

Why is the infinite sum of this trigonometric series =zero yet when I evaluate the finite sum and take the limit $n \to \infty$ it not well defined?

Evaluating the infinite sum of the below trigonometric series yields a result of zero. However, when I calculate the finite sum up to $n$ and subsequently take the limit as $n$ approaches infinity on ...
ivan44's user avatar
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3 votes
0 answers
127 views

Calculate $\sum\limits_{k=1}^{\infty} \frac{\sin 2k}{3^k}$

I have solved it, but it does not match with the last part of solution. The logic is: Let's consider complex series $\sum\limits_{k=1}^{\infty} \frac{\cos 2k + i\sin 2k}{3^k}$, imaginary part of which ...
bag_dush's user avatar
  • 105
3 votes
1 answer
86 views

Question about arcsec, trig sub and absolute value of $|x|$.

$\newcommand{\arcsec}{\operatorname{arcsec}}$I was solving this integral recently when I was comparing my answer to another answer in desmos and mine was slightly off but I fixed it by wrapping part ...
Dryden Bryson's user avatar
1 vote
2 answers
45 views

General approaches to solve trigonometric polynomial equations

Assume we are given an arbitrary trigonometric polynomial equation $$ f(\theta)=\sum^{n}_k(a_k cos(k\theta)+b_k sin(k\theta))=0, \ \ $$ where $\theta\in[0, 2\pi)$ and $a_k, b_k$ is real . I would ...
Yunzhe's user avatar
  • 133
1 vote
1 answer
98 views

Does $ \sum_{n=-\infty}^{\infty}{{1}\over{x-\pi n}} $ converge to cotangent?

I was experimenting and I found the sum $$ \sum_{n=-\infty}^{\infty}{{1}\over{x-\pi n}} $$ As I input higher values for "infinity" (the graphing calculator I'm using doesn't let you put ...
Jme's user avatar
  • 83
1 vote
0 answers
58 views

Deduce that $\sum_{n=1}^\infty\frac{1}{1+n^2}=\frac{1}{2}(\pi\coth\pi-1)\quad$ [duplicate]

I am having problems showing that $$\sum_{n=1}^\infty\frac{1}{1+n^2}=\frac{1}{2}(\pi\coth\pi-1)\quad $$ Here's my attept to this point: I tried to express each term using a partial fraction ...
Bagaringa's user avatar
  • 402
1 vote
0 answers
45 views

Fejer-Riesz Theorem for analytic polynomials

Let $\mathbb{D}=\{z: |z| <1\}$ and $\mathbb{T}=\{z: |z|=1\}$. Suppose $D$ and $E$ are polynomials of degree atmost $n$ with complex coefficients such that $|E(z)| \leq |D(z)|$ for all $z \in \...
Curious's user avatar
  • 933
4 votes
2 answers
158 views

Can we find the exact value of a double sum with cosine without differentiation?

After finding an interesting double sum $$\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{(-1)^{m+n}}{(m+n)^2} = \frac{\pi^2}{12}-\ln 2 ,$$ I started to investigate a harder one $$\displaystyle \sum_{m=...
Lai's user avatar
  • 21.2k
1 vote
3 answers
174 views

Show $8 \sin \frac{4 \pi}{9} \sin \frac{2 \pi}{9} \sin \frac{\pi}{9} = \sqrt{3}$.

Please provide guidance on how to solve the product to sum question. I have also attached my attempt which was unsuccessful... Show $8 \sin \frac{4 \pi}{9} \sin \frac{2 \pi}{9} \sin \frac{\pi}{9} = \...
racer234's user avatar
2 votes
1 answer
89 views

Some interesting trigonometric sums

In working on a physics problem, I've come across sums of trigonometric functions of the following form: $$S(n,L) = -4^{n}+2^{2n}\sum_{k=0}^{L}\left[\cos\left(\frac{k\pi}{L+1}\right)\right]^{2n}$$ ...
miggle's user avatar
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1 vote
0 answers
44 views

Use complex numbers to simplify this trigonometric expression [duplicate]

I was able to simplify this expression using transformation formulae. The answer turned out to be $$\frac{\sqrt{7}}{2}=\sin\left(\frac{2\pi}{7}\right)+\sin\left(\frac{4\pi}{7}\right)+ \sin\left(\frac{...
Rexquiem's user avatar
  • 334
3 votes
1 answer
95 views

About $f(x) = \frac{\sum_{n=1}^{\infty} \sin^2(x/n)}{x}$

Consider for $x>0$ $$f(x) = \frac{\sum_{n=1}^{\infty} \sin^2(x/n)}{x}$$ I was fascinated by the behaviour of this function. It is easy to show that $$\lim_{x \to +0} \frac{f(x)}{x} = \zeta(2) = \...
mick's user avatar
  • 16.1k
2 votes
1 answer
183 views

Summation inside tan(x)

Given $ a_1 +a_2 + a_3+...+a_n= \theta$ degrees. Where $tan(a_k) = \frac{n}{n^2 + k(k-1)}$. Find $tan(\theta)$ in terms of "n". I tried using the formula tan(a+b+c+d+...) = $\frac{S_1-S_3-...
BlackHood's user avatar
  • 171
5 votes
1 answer
109 views

$\lim_{p\to\infty}\sum_{n≥1}\arctan\left(\frac{1}{n^p}\right)=\frac{\pi}{4}$

I was solving simple series involving the function $\arctan(x)$ of the form : $$\sum \arctan\left(\frac{d}{ax^2+bx+c}\right)$$ where $a,d,b,c$ are constants. This series can be manipulated too by ...
An_Elephant's user avatar
  • 2,756
1 vote
1 answer
57 views

Representation of Weierstraß $\wp$ function for $\Lambda=\Bbb Z+\Bbb Z i$ as series over trigonometric function

The Weierstraß $\wp$ function for a lattice $\Lambda\subset\Bbb C$ can be defined by the sum $$ \wp(z) = \frac1{z^2} ~+\!\! \sum_{\lambda\in\Lambda\setminus\{0\}} \left(\frac1{(z-\lambda)^2}-\frac1{\...
emacs drives me nuts's user avatar
1 vote
2 answers
131 views

How to find the summation of the above trigonometric series without using desmos?

How to find the summation of $$ \sin(2 + \sin(2 + \sin(2 + \cdots \infty)))? $$ I am trying this question by denoting the above summation as $S$. Therefore, $$ S = \sin(2 + \sin(2 + \sin(2 + \cdots \...
Syamaprasad Chakrabarti's user avatar
3 votes
1 answer
93 views

How to find the summation of the following inverse trigonometric series?

How to find $$\sum_{i=1}^{n} \sin^{-1}\left(\frac{1}{i(i+1)}\right)?$$ I am trying this question by writing $1$ as $(i+1)-i$. So, the above summation will become $$\sum_{i=1}^{n} \sin^{-1}\left(\frac{(...
Syamaprasad Chakrabarti's user avatar
0 votes
1 answer
62 views

How to evaluate the following trigonometric sum?

I recently learned summing a trigonometric series where the $i^{th}$ term is $\cos(i\theta)$ where the angles are in arithmetic progression. The idea is to multiply the sum $S$ by $\sin\theta$ and ...
RajaKrishnappa's user avatar
0 votes
1 answer
56 views

Trigonometric sum related to ellipse

How to prove this identity for $2\le n\in\Bbb N^+, a , b > 0$? $$ \sum_{k=0}^{n-1}\frac1{a^2\cos^2\left(\frac kn⋅π\right)+b^2\sin^2\left(\frac kn⋅π\right)}=\frac{n}{ab}⋅\frac{(a+b)^n+(b-a)^n}{(a+b)...
hbghlyj's user avatar
  • 2,770
0 votes
1 answer
59 views

$\lim_{n\rightarrow\infty} \sum_{k=1}^n \text{arccot}(2k^2)$

I think this can be solved using the sandwiching theorem but I have not been able to find appropriate series to sandwich with. The trivial substitution yields that the limit is between $0$ and $\infty$...
Marin's user avatar
  • 187
0 votes
0 answers
33 views

Prove a suspected equality between an infinite sum of sinc-like terms and an infinite sum of cosines

I stumbled on the following suspected equality of infinite sums that I believe holds for $c \in \mathbb{R}$ and $x \in \mathbb{R} - \mathbb{Z}$: $$\sum_{n=-\infty}^\infty \frac{\sin(2 \pi c (x - n))}{\...
crb233's user avatar
  • 1,022
0 votes
0 answers
79 views

How to prove this inequality about trigonometric polynomial?

Problem statement: Define $T_n(x)=\frac{a_0}{2}+\sum_{k=1}^{n}{(a_k\cos kx+b_k\sin kx)}$ be a real valued trigonometric polynomial on $[-\pi,\pi]$.Prove that $$\mathop{max}\limits_{-\pi\le x \le \pi}{|...
kmxzc's user avatar
  • 21
-1 votes
3 answers
72 views

trigonometric way to solve Sine and Cosine sums? [duplicate]

drawing graphs for most addition or subtractions of Sine and Cosine formula lead to another sine or cosine shaped graph, but I don't know how to actually write them as a formula. For example , drawing ...
Aug's user avatar
  • 25
0 votes
1 answer
85 views

Solution for $x$, $\sin^{-1} (x-\frac{x^2}{2}+\frac{x^3}{4}-\cdots)+\cos^{-1} (x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots)=\frac{\pi}{2}$

Solution for $x$, $\sin^{-1} (x-\frac{x^2}{2}+\frac{x^3}{4}-\cdots)+\cos^{-1} (x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots)=\frac{\pi}{2}$ I have tried, $ 1-\frac{x^2}{2}+\frac{x^3}{4}-\cdots=:y$ Now, ...
O M's user avatar
  • 2,134
1 vote
1 answer
139 views

Prove $\sum_{n≥1}\frac{\sin n}{n}= \frac{1}{2}(\pi-1)$

I recently learnt that sums of sine and cosine series whose arguments are in arithmetic progression can be evaluated using complex numbers. I modified it a little; like below I divided it by the ...
An_Elephant's user avatar
  • 2,756
0 votes
0 answers
22 views

Weak L^2 Zeros. Cone-Like Zeros.

For sake of notation, let $Q = [0,1]\times [0,1]$. Additionally, define $$ \rho_{x_0,\xi_0}(x,\xi) = \sqrt{(x-x_0)^2+(\xi-\xi_0)^2}. $$ Definition. Let $g\in L^2(Q)$. We say that the point $(x_0, \...
Doofenshmert's user avatar
0 votes
1 answer
66 views

Prove that Definite Integral of Function of Cos (t) is Larger than Definite Integral of same Function multiplied by Sin (t)

Can you please clarify how to prove this? $$\int_{0}^{\pi}\left(\frac{1+\cos t}{2}\right)^{k}\operatorname{d}t>\int_{0}^{\pi}\left(\frac{1+\cos t}{2}\right)^{k}\sin t\operatorname{d}t$$ Here, $k=1,...
texmex's user avatar
  • 800
2 votes
2 answers
166 views

Generalized formula for $\sin((2n-1)x)$?

Does there exist a generalized formula for $\sin((2n-1)x)$? I noticed that if, $\sin(1x)=t^1$ Then $\sin(3x)=3t^1-4t^3$ $\sin(5x)=5t^1-20t^3+16t^5$ $\sin(7x)=7t^1-56t^3+112t^5-64t^7$ $\cdots$ They do ...
user avatar
-1 votes
1 answer
92 views

Prove That an Approximation of $\sin(x)$ via Euler's Formula Approaches $\sin(x)$

I'm trying to approximate the trigonometric functions for a code library, and I want to ask if this is a good way to go about it. I'm aware of the Taylor series approach, but I wanted to go with ...
Sig Moid's user avatar
10 votes
2 answers
296 views

Find the value of $\dfrac{\cos(\pi/4)\cos(\pi/6)\cos(\pi/8)\cos(\pi/10)\cos(\pi/12)\cdots}{\cos(\pi/3)\cos(\pi/5)\cos(\pi/7)\cos(\pi/9)\cdots}$

I manually calculated the values and found that the resulting answer was very close to $1.57\simeq \dfrac{\pi}{2}$ $$\dfrac{\cos(\pi/4)\cos(\pi/6)\cos(\pi/8)\cos(\pi/10)\cos(\pi/12)\cdots}{\cos(\pi/3)\...
user avatar
1 vote
1 answer
61 views

Simplifying $\sin(nx)$ series notation

I am attempting to simplify the following notation: $$\sin(nx)=\sin(x)\left(2^{n-1}\cos^{n-1}(x)-\frac{n-2}{1!}2^{n-3}\cos^{n-3}(x)+\frac{(n-2)(n-3)2^{n-5}\cos^{n-5}(x)}{2!}+\cdots\right)$$ I ...
joe_bill.dollar's user avatar
16 votes
2 answers
654 views

Simplifying $3S_1 + 2S_2 + 2S_3$, where $S_1=2\sum_{k=0}^n16^k\tan^4{2^kx}$, $S_2=4\sum_{k=0}^n16^k\tan^2{2^kx}$, $S_3=\sum_{k=0}^n16^k$

If $$S_1=2\sum_{k=0}^n 16^k \tan^4 {2^k x} $$ $$S_2=4\sum_{k=0}^n 16^k \tan^2 {2^k x} $$ $$S_3= \sum_{k=0}^n 16^k $$ Find $(3S_1 + 2S_2 + 2S_3)$ as a function of $x$ and $n.$ In the expression asked ...
Maths's user avatar
  • 495
1 vote
0 answers
42 views

Evaluate the product $\prod\limits_{k=1}^{2^{1999}} (4\sin^2(\frac{k\pi}{2^{2000}})-3)$ [duplicate]

Evaluate the product $$\prod\limits_{k=1}^{2^{1999}} \left(4\sin^2\left(\frac{k\pi}{2^{2000}}\right)-3\right)$$ I tried by making it into $\frac{\sin 3\theta}{\sin\theta}$ form but couldn’t proceed ...
Maths's user avatar
  • 31
3 votes
0 answers
85 views

How can i prove this sum of factorials?

i was messing around whith some trig identities and i came across this equation: $$\sum_{k=0}^n\frac{1}{(2k)!(2(n-k)+1)!}=\frac{1}{2}\frac{2^{2n+1}}{(2n+1)!}\quad\quad(1)$$ This formula becomes pretty ...
alberto mazzarotto's user avatar
6 votes
1 answer
94 views

Sum of the Sines of the Ratios of Fibonnaci Numbers

Saw an online a proof that $$\lim_{n\to \infty}\sum_{k=1}^{n}\sin\left(\frac{k}{n^2}\right)=\frac{1}{2}$$ that utilized the Squeeze Theorem and the fact that $x-x^3\leq \sin(x)\leq x$ for small $x$. ...
Mauithedog10's user avatar
2 votes
4 answers
103 views

Prove $\sum_{k=0}^{\infty}a^k\text{cos}kx=\frac{1-a\text{cos}x}{1-2a\text{cos}x+a^2}, \; |a|<1$

I want to prove the following; $$\sum_{k=0}^{\infty}a^k\cos{kx}=\frac{1-a\cos{x}}{1-2a\cos{x}+a^2}, \; |a|<1 \tag{1}$$ I know that; $$\text{S}(r)=\sum_{k=0}^{\infty}r^k=\frac{1}{1-r}, \; |r|<1$$ ...
Rasmus Andersen's user avatar
0 votes
1 answer
40 views

Loss of convergence at end point

Recently i have been reading "A radical approach to real analysis" by David Bressoud and this thing got stuck in my mind Here you might lose convergence at the end point is written in last ...
Feather's user avatar
  • 41
10 votes
1 answer
384 views

Is there an identity for $\sum_{k=0}^{n-1}\csc(w+ k \frac{\pi}{n})\csc(x+ k \frac{\pi}{n})\csc(y+ k \frac{\pi}{n})\csc(z+ k \frac{\pi}{n})$?

What I'd like to find is an identity for $$\sum_{k=0}^{n-1}\csc\left(w+ k \frac{\pi}{n}\right)\csc\left(x+ k \frac{\pi}{n}\right)\csc\left(y+ k \frac{\pi}{n}\right)\csc\left(z+ k \frac{\pi}{n}\right)$$...
onepound's user avatar
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