Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange
Join us in building a kind, collaborative learning community via our updated Code of Conduct.

Questions tagged [trigonometric-series]

For questions about or related to trigonometric series.

0
votes
1answer
35 views

Closed form for a conditional trigonometric series

Do you know an easy way to prove the following $\forall~L$? $\sum\limits_{\scriptstyle k = 1,~k \ne L\atop \scriptstyle ~l = 1,~l \ne L}^N {\cos \left( {\frac{{2\pi }}{N}n(k - l)} \right)} - 2\sum\...
1
vote
1answer
36 views

Summation of Finite trigonometrical series

Let $O$ be any point on the circumference of a circle circumscribing a regular polygon $A_1,A_2,A_3.., A_{2n+1}$ such that $O$ lies on the arc $ A_1A_{2n+1} $. Show that $OA_1+OA_3+...OA_{2n+1}=OA_2+...
3
votes
1answer
97 views

Any suggestions on how to compute $\limsup |\cos n|^{n^2}$?

This problem has proven very difficult, does anyone have any suggestions on how to tackle it? Any little known theorems/identities that might help?
1
vote
0answers
27 views

$\sum_{k=1}^n\sin kx=\frac{\cos\left(\frac{1}2x\right)-\cos\left(n+\frac{1}2\right)x}{2\sin(x/2)},x\neq0,\pm\pi,\pm 2\pi,\ldots$ [duplicate]

Maybe I have a trivial question but , Could you tell me how did we get this fact please ? $$\sum_{k=1}^n\sin kx=\frac{\cos\left(\frac{1}2x\right)-\cos\left(n+\frac{1}2\right)x}{2\sin(x/2)},x\neq0,\pm\...
2
votes
1answer
69 views

Expansion of $x$ in powers of $u$

Given: $$\sin(x) = u \sin(x+a),\qquad {u<1}$$ How do I expand $x$ in powers of $u$? I tried using Taylor series but it failed to proceed.
3
votes
0answers
144 views

Extended conjecture for $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin\left(\sum\limits_{k = 1 }^\infty \frac{a_k \pi}{b_k}P_k(n) \right)$

I asked a question that is related to this question and claimed that Generalized Conjecture: $$f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin\left(P_r(n) \frac{a \pi}{b}\right) \tag 1 $$ I have a ...
2
votes
3answers
284 views

Conjecture about $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin\left(P_r(n) \frac{a \pi}{b}\right) $

I asked another question related this question. $r=1$ was considered in the related question.You may see proofs for $r=1$. I would like to generalize the conjecture when $r$ is any positive integer ...
1
vote
0answers
60 views

Number of solutions $2^\frac{1}{\sin^2 x_2} \cdot 3^\frac{1}{\sin^2 x_3}\cdot\;\cdots\;\cdot n^\frac{1}{\sin^2 x_n} \leq n!$ with $x_i\in(0,4\pi)$

There was one question in a question set that I was attempting that went something like this: The number of solutions of the following inequality $$2^\frac{1}{\sin^2 x_2} \cdot 3^\frac{1}{\sin^2 x_3} \...
1
vote
1answer
29 views

Polar Curve Fitting

I am working on automated counting and one of my solutions is the use of the template matching algorithm (specifically using Chamfer Matching Algorithm). However, granted it is a template matching ...
0
votes
0answers
41 views

How to convert my equation (Exponential) to cot form?

I have the below equation: $$ F(t)=(k_1 e^{at} + (k_{Re}-ik_{Im})(\lambda+i\omega)e^{(\lambda+i\omega)t}+(k_{Re}+ik_{Im})(\lambda-i\omega)e^{(\lambda-i\omega)t})/((r_1k_1 e^{at} + r_2(k_{Re}-ik_{Im})(...
13
votes
3answers
544 views

Prove or disprove that $ \sum\limits_{k = 1 }^T f(k)=0 $ where $f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin(\frac{n(n+1)(2n+1)}{6}x) $

$$f(m)=\sum\limits_{n = 1 }^ m (-1)^n \sin\left(\frac{n(n+1)(2n+1)}{6} \frac{a \pi}{b}\right) \tag 1 $$ Where $a,b,m$ positive integers. I have tested in WolframAlpha for many $a$ and $b$ values. I ...
1
vote
1answer
68 views

how to get $\sum_{k=1}^\infty \arctan\biggr(\frac{10k}{(3k^2+2)(9k^2-1)}\biggr)=\log3-\frac{\pi}{4}$

problem in the above asked equation of S.Ramanujan ! Hello everyone,this is a result of an entry described by ramanujan,i first request you to see the photo i have attached. click here for the image ...
2
votes
0answers
46 views

Trouble with infinite summation involving $\sin$ function

How can I proceed for this: $$\sum_{n=-\infty}^{+\infty}{\bigg(\frac{\sin{(an+b)}}{an+b}\bigg)^2}$$ I know the answer is $\frac{\pi}{a}$, but how? I think it could be related to the Shannon sampling ...
0
votes
1answer
32 views

Let $f(t)$ be a polynomial. When $\sum_{k=1}^n f(\sin{kx_0})$ is bounded for any $x_0$?

Let $x_0 > 0$, $f(t)$ be a polynomial. What is condition specified for $f(t)$ to sequence $(s_n)=\sum_{k=1}^n f(\sin{kx_0})$ be bounded? For example: if $f(t) = t$ then $|(s_n)|=|\sum_{k=1}^n \sin{...
0
votes
1answer
44 views

Arcsin Series : $\sum\limits_{n=1}^{\infty}\Bigl [\frac {\pi}{2} -\arcsin\bigl(\frac{n}{n+1}\bigr)\Bigr]^{\alpha}$

Good morning everyone, I'd like to discuss with you the following exercise : $$\sum\limits_{n=1}^{\infty}\Bigl[\frac {\pi}{2} - \arcsin\Bigl(\frac{n}{n+1}\Bigr)\Bigr]^{\alpha}$$ After verified the ...
0
votes
1answer
66 views

I found a weird occurrence with equal angle polygons and sine waves and i need help proving it

Here is a desmos graph that visualizes what I am about to say Okay, let's say we have a polygon with $s$ sides and $a = \frac{360°}{s}$. All of those polygon's angles are equal and all of it's sides ...
4
votes
1answer
44 views

Why is the Taylor Series of tan⁻¹x valid for x=1?

For -1 < x < 1, $$\dfrac{1}{1-x} = 1+x+x²+x³+...$$ Then $$\frac{1}{1-(-x²)}= 1-x²+x⁴-x⁶+...$$ Integrating both sides with respect to x from 0 to x we obtain, tan⁻¹x = $$x - \dfrac{x³}{³} + \...
0
votes
2answers
80 views

Make $\sin(x)-x\cos(x)$ beautiful?

When computing a Fourier series I came across a term like $$\sin(x)-x\cos(x)$$ Is there a way to reduce this expression, e.g. to only $sin$ or $cos$? My final series looks like this: $$ f(t) = \dfrac{...
1
vote
0answers
24 views

Regarding Generalized Products of Cosines

I was looking at this identity (see below) from Wikipedia, and I don't quite understand how you can add and multiply elements from $S$. Elements of $S$ are sequences of -1's and 1's, so how are ...
1
vote
1answer
49 views

Evaluating $\cos\frac{k\pi}{p}+\cos\frac{2k\pi}{p}+\cos\frac{3k\pi}{p}+\dots +\cos\frac{2\frac{(p - 1)}{2}k\pi}{p}$ [duplicate]

If $p$ is a prime then what is the value of the series $$\cos\frac{2\pi}{p}+\cos\frac{4\pi}{p}+\cos\frac{6\pi}{p}+\dots +\cos\frac{(p - 1)\pi}{p}$$ In general what is the value of the following ...
1
vote
2answers
71 views

Find the value of $\cot(16)\cot(44)+\cot(44)\cot(76)-\cot(76)\cot(16)$

Find the value of $$S=\cot(16)\cot(44)+\cot(44)\cot(76)-\cot(76)\cot(16)$$ Note:All angles are in degrees My method: I used the identity $$\tan(x)\tan(60+x)\tan(60-x)=\tan(3x)$$ So choosing $x=16$...
1
vote
1answer
115 views

Is there a good estimate for $\sum_n\frac{1}{\cos(n\theta)}$?

Is there a good estimate for the following series (when $\theta$ is very small)? $$\sum_{n=1}^{\lfloor\frac{\pi}{4\theta}\rfloor}\frac{1}{\cos(n\theta)}$$ My original problem is: Given a right ...
0
votes
0answers
42 views

Show that the sum of a convergent series is odd

Let $ b_k = \frac{1}{k(\ln(1+k))^2} $ for all $k \in \mathbb{N}$. Consider the following convergent trigonometric series: $$ \sum_{k=1}^{\infty} b_k \sin(kx) $$ Show that the sum is odd. Here ...
0
votes
0answers
24 views

Write trigionemtric series on exponential form

I have $b_k=\frac{1}{k(ln[1+k])^2}$ for all $k\in\mathbb{N}$. We have the triginometric series: $$\sum_{k=1}^{\infty}b_k\sin(kx),$$ and has to show that the sum is odd, and write on exponential form. ...
0
votes
2answers
58 views

Interval of convergence of trig. series involving complex numbers

I have been working on the following problem: Determine the sum of the convergent trigonometric series: $$ \sum_{k=-\infty}^{\infty} \frac{i}{3^{|k|}}e^{ikx} $$ This is my work so far: Because ...
0
votes
1answer
26 views

Integration of Fourier series produces denominator of $0$

I want to explore the integral of the Fourier series for an impulse train: $$\sum_{k=0}^R \frac{e^\frac{2 i k \pi x}{R+1}}{R+1}$$ where $i=\sqrt(-1)$. I find $$\int \sum_{k=0}^R \frac{e^\frac{2 i ...
0
votes
0answers
14 views

Image of the Trigonometric basis

Let $\varphi_{2k} : x \mapsto \cos(2\pi kx)$ and $\varphi_{2k+1} : x \mapsto \sin(2\pi kx) $ denote the (not normalized) trigonometric basis. Is there anything that can be said about the set : $\...
4
votes
1answer
60 views

Evaluating $\tan\left(\sum_{r=1}^{\infty} \arctan\left(\frac{4}{4r^2 +3}\right)\right)$ [duplicate]

$$\tan\left(\sum_{r=1}^{\infty} \arctan\left(\dfrac{4}{4r^2 +3}\right)\right)= ? $$ I wrote it in the form: $$\tan\left(\sum_{r=1}^{\infty} \arctan\left(\dfrac{\dfrac43}{\dfrac{4r^2}{3} +1}\right)...
3
votes
1answer
27 views

Proving the formula of a function much similar to the Dirichlet kernel.

Many of us know about the Dirichlet kernel which is on lower level stated as $$D_n(\theta) =\frac {\sin \left(n+\frac 12\right)\theta}{\sin \frac {\theta}{2}}=1+2\sum_{r=1}^n \cos (r\theta)$$ I ...
2
votes
1answer
65 views

Prove that $\tan^{-1}(1/n)+\tan^{-1}(2/n)+\cdots+\tan^{-1}(n/n)$ increases as $n$ increases

Let $$f(n)=\tan^{-1}\left(\frac{1}{n}\right)+\tan^{-1}\left(\frac{2}{n}\right)+\tan^{-1}\left(\frac{3}{n}\right)+ \cdots +\tan^{-1}\left(\frac{n}{n}\right)$$ where $n\in\mathbb{N}$. Prove ...
-1
votes
2answers
40 views

Sum of sines inequality [closed]

I need to prove the following inequality: $$\bigg\lvert \sum_{n=1}^{N}\sin(nx)\bigg\rvert \leq \frac{1}{\sin(x/2)}, \, x\neq 2k\pi,k\in \mathbb{Z}$$ No idea where to start. Any tips?
0
votes
1answer
65 views

$\frac{1}{n}-\sin(\frac{1}{n})\sim\frac{1}{6n^3}$?

I saw this approximation posted in a different thread Convergence of the series $\sum_{n=1}^{\infty}((1/n)-\sin(1/n))$..?. I am very curious as to how one would come up with this approximation. My ...
1
vote
3answers
61 views

Limiting Behavior of the oscillating series

Consider the function $$ f(x):=\sum_{n=1}^\infty \frac {\sin (x/2^n)}{2^n},\quad x\in \mathbb R. $$ Is it true that $\lim_{x\to \infty}f(x)=0$? The series converges uniformly and absolutely on $\...
1
vote
1answer
116 views

Sine series: angle multipliers add to 1

It is known that in an sine series with angles in arithmetic progression (I refer to this question): $\sum_{k=0}^{n-1}\sin (a+k \cdot d)=\frac{\sin(n \times \frac{d}{2})}{\sin ( \frac{d}{2} )} \times ...
0
votes
0answers
34 views

Equivalence between $\phi (z,s,a)$ and a sum of single impulses

A summed set of (negative) single impulses is given by $-\sum_{R=0}^m \frac{\sin \pi (x-(R+1))}{\pi (x-(R+1))}$ Mathematica simplifies this to a function involving the Lerch Transcendent $\phi (z,s,...
1
vote
1answer
27 views

Trig Sequence Puzzel?

Let $a_0 = \sqrt2 + \sqrt 3 + \sqrt 6$ and let $a_{n+1}=\frac{(a_n)^2-5}{2(a_n)+2}$ for $n\ge0$. Prove that $a_n=\cot\left(\frac{2^{n-3}\pi}{3}\right)-2$ for all n. I have already proved that $a_0 = \...
0
votes
1answer
53 views

Subtle Carefulness in use of AM-GM inequality

The question :- The solution given in textbook :- My answer :- I am not able to identify which is right. I thought mine may be wrong due to squaring or reciprocal taking but I don't think it is ...
1
vote
1answer
67 views

Does this formula for a sum of cosines has a name?

On this thread, Bernard kindly gave me the formula $$\sum_{k=0}^R \cos k \theta = \frac{\sin \frac{(R+1)\theta}{2}}{\sin \frac{\theta}{2}} \cos \frac{R \theta}{2}$$ He describes the formula as well-...
1
vote
3answers
72 views

Find the value of $\sin(\tfrac{\pi}3) + \tfrac12\sin(\tfrac{2\pi}3) + \tfrac13\sin(\tfrac{3\pi}3)+\dots$ up to infinity

Initially, I thought of doing this by first evaluating the function f(x) = cos(x) + cos(2x) + cos(3x) +..... and then integrating it. However, I cant seem to find a proper integratable function (if ...
3
votes
3answers
73 views

Why are these $\sum \cos$ and $\csc$ equivalent?

Mathematica 'simplifies' this formula $$\sum_{k=1}^R \cos \frac{2k \pi x}{R}$$ to this $$\frac{1}{2} \biggl(\csc \frac{\pi x}{R} \sin \frac{(2R+1) \pi x}{R}-1\biggr)$$ A graphical plot of the two ...
4
votes
5answers
87 views

For rotation matrix $A$, find $B = A^4- A^3 + A^2 - A$

Find the value of $B = A^4- A^3 + A^2 - A$ where $A$ is the matrix given below $$ A= \left [ \begin{matrix} \cos\alpha & \sin \alpha \\ -\sin\alpha & \cos\alpha \end{matrix} \...
2
votes
0answers
60 views

Proving a general formula for $\sum_{m=0}^{n}(-1)^m\binom{n}{m} \sin(a+(n-m)h)$

While solving some of limits questions I noticed a very remarkable formula popping up which I think is pretty interesting. But I am not able to prove the formula with general values of $n$. The ...
0
votes
1answer
30 views

Convergent series with trigonometric functions like $a_{0}=m, a_{2n-1}=\sin(a_{2n-2}), a_{2n}=\cos(a_{2n-1})$

I have found by computation that $$a_{0}=m, a_{2n-1}=f(a_{2n-2}), a_{2n}=g(a_{2n-1})$$ converges when $m$ - real and $f,g$ - some trigonometric function. $$(f,g)\to(|\lim\limits_{n\to\infty}a_{2n-1}|, ...
0
votes
1answer
49 views

Simplify $\sin(\alpha)+\sin(\alpha+x)+\sin(\alpha+2x)+\sin(\alpha+3x)+ \dotsb +\sin(\alpha+nx)$ [duplicate]

$n$ is known, $\alpha$ is irrelevant, and you are looking for $x$? How do you simplify: $$\sin(\alpha)+\sin(\alpha+x)+\sin(\alpha+2x)+\sin(\alpha+3x)+ \dotsb +\sin(\alpha+nx)$$
0
votes
0answers
33 views

Finding closed forms for a trigonometric series or for the partial sums

Is there any closed formula for the series $$\sum_{k=1}^{\infty}\frac{\sin (kx)}{k}$$ or for the sum $$\sum_{k=1}^{n}\frac{\sin (kx)}{k}$$ where $x$ is an real number. Thank you.
2
votes
1answer
56 views

Formula for summing an arbitrary number $n$ of cos functions?

This post gives $$\cos A+\cos B+\cos C=1+4\sin \frac {A}{2}\sin \frac {B}{2}\sin \frac {C}{2}$$ Is it possible to derive a generalised formula for $$\cos A+\cos B+\cos C+...+\cos N$$ i.e., a ...
1
vote
1answer
48 views

Limit of the infinity product goes to zero

Does anyone know if there is a value of $z$ between $0$ and $\pi$, such that the product below equals zero? $$ \lim_{m \rightarrow \infty} \prod_{k=1}^{m}\left(1-\frac{z^2}{4k^2\pi^2}\right) \prod_{k=...
0
votes
1answer
50 views

solve for x within a summation equation

I am Only in Algebra II Trig and am just getting into these kinds of things. So please forgive me if my terminology or notation are wrong. So, Say I have $$\sum_{n=0}^\infty \frac{(-1)^{n} x^{2n+1}}{(...
20
votes
3answers
2k views

Consecutively Adding Sines

One thing, I'm not a mathematician so please be patient. I am still in Algebra II Trig. Leading with that, why does $$ x_0 = \sin 1, \space x_1 = x_0 + \sin x_0, \space x_2 = x_1 + \sin x_1 ... $$ ...
1
vote
2answers
89 views

A very challenging question integral of an infinite product.

Evaluate: $$\int_{0}^{\infty}\sqrt{\prod_{k=0}^{\infty}\frac{\cos(θ/2^k)+1}{2}}dθ.$$ This was a problem from the book 'S.P.Patterson's Rectreational Problems in Advanced Mathematics'. The problem was ...