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Questions tagged [trigonometric-series]

For questions about or related to trigonometric series.

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13 views

Simpson vs. trapezoidal rule for numerically integrating cos(x)*cosh(x) in range 0 to pi?

I have to numerically calculate many integrals similar to this: $$\int_0^\pi \cosh{\left(\frac{a_1\cos{x}+a_2\cos{2x}+a_3\cos{3x}+\ldots}{10}\right)}\cos{jx}\cos{kx}\space dx$$ Right now I am using ...
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1answer
26 views

General term for the sum $\sum \sin(k)$ [duplicate]

How do I prove that: $$\sin(1)+\sin(2)+\cdots+\sin(n)=\frac{\sin\left(\frac{n+1}2\right)\sin\left(\frac n2\right)}{\sin\left(\frac12\right)}?$$ I have tried to to use the formula $\sin(2x)=2\sin(x)\...
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1answer
27 views

Approximate Trig Functions without the use of Taylor Series

I am familiar with how a trig function, i.e. $\sin(x)$, can be approximated by a MacLauren series; \begin{align} \sin(x_0) &\approx \sin(0) + \cos(0) x_0 - \frac{1}{2}\sin(0) x_0^2 - \frac{1}{3!}\...
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0answers
41 views

On limit of sum $\lim\limits_{s\to0^+}\sum\cos\left(\pi\frac{n}{m}\right)/{n^s}$ & $\lim\limits_{s\to0^+}\sum\sin\left(\pi\frac{n}{m}\right)/{n^s}$

$(1).$ Show that: $$ \lim_{s\to0^+}\,\left[\sum_{n=1}^{\infty}\cos\left(\pi\frac{n}{m}\right)\frac{1}{n^s}\right]=\color{red}{-\frac{1}{2}} \quad\colon\space\forall\,m\in\mathbb{N}^{+}\tag{1} $$ ...
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0answers
16 views

ratio test to prove trigonometric series converges

I'm trying to convince myself that the following trigonometric series converges for all values of x $\sum_{n=1}^\infty \dfrac{sin(nx)+cos(nx)}{n^2}$ my approach to the ratio test was following: $\...
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3answers
122 views

Limit of a sum using complex analysis.

I'm trying to find the limit of this sum: $$S_n =\frac{1}{n}\left(\frac{1}{2}+\sum_{k=1}^{n}\cos(kx)\right)$$ I tried to find a formula for the inner sum first and I ended up getting zero as an answer....
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0answers
26 views

Find minimum maximum of sum of absolute values of sines, offset by equidistant phases

For every integer $N>0$ given function $f_N(x) = \sum_0^{N-1} |\sin(x+\frac{2i\pi}{N})|$ Is there some $O(1)$ analytic solution (without using $\sum$ operation), to find its minimum $\min(f_N)=?$ ...
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1answer
78 views

Convergency of $\sum_{n=1}^{\infty}\frac{\csc(n)}{n!}$

I am stuck on how to prove the convergency of the series $$\sum_{n=1}^{\infty}\frac{\csc(n)}{n!}.$$ It seems like that the series converges to approximately $2.85$, but I have no idea how to show ...
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2answers
59 views

How do you evaluate this trigonometric sum?

I have strong reason$^{\dagger}$ to believe that the following equation is true: $$\sum_{m=0}^{n} \left[\left(e^{i\pi\frac{k+k'}{n}}\right)^m+\left(e^{i\pi\frac{k-k'}{n}}\right)^m+\left(e^{i\pi\frac{-...
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1answer
55 views

Partial sums of $\frac{\pi}{2}=\sum_{n=0}^\infty \frac{(2n-1)!!}{2^n\cdot n!\cdot (2n+1) }$.

Recently, I have found a formula for $\pi$. That is $$\frac{\pi}{2}=\sum_{n=0}^\infty \frac{(2n-1)!!}{2^n\cdot n!\cdot (2n+1) }$$ However, the problem arises when you take the partial sums. For ...
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2answers
140 views

This has to be in the lit somewhere. Can someone point me to this in any accessible book or lit?

It's just a big trig, sinusoidal, Fourier series thing: $$\begin{align} y(t) &= \sum_{k=0}^{K} a_k \big( A \cos(\omega t) \big)^k \\ \\ &= \sum_{n=0}^{K} b_n \cos(n \omega t) \\ \end{...
3
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1answer
50 views

Where should I find the ranges for $\sum_{n=1}^k \sin n$ and other similar trigonometric series?

It can be found that $$\sum_{n=1}^k \sin n = \frac{\sin\left(\frac{k+1}{2}\right)\sin\left(\frac{k}{2} \right)}{\sin\left(\frac{1}{2}\right)},$$ $$ \sum_{n=1}^k \cos n = \frac{\cos\left(\frac{k+1}{...
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2answers
93 views

When does the limit $|\cos(n)|^{f(n)}$ converges as $n \rightarrow \infty, n \in \mathbb{N}$?

Here we go with a not-so-trivial problem: Inspired by another problem that I myself asked here, I came with this more general formulation: Let be the sequence $a(n) = |\cos(n)|^{f(n)}$. Then, when ...
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2answers
117 views

Does the limit of $\cos^{2n}(n)$; $n$ a positive integer; converge as $n\to\infty$?

I'm struggling with what it seems to be a pretty simple limit: $$\lim_{n \rightarrow \infty} \cos^{2n}(n)$$ I have arguments to believe that this limit converges to $0$ because $n \in (kπ, (k+1)π) $ ...
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2answers
200 views

Sum to n terms the series $\cos \theta+ 2\cos 2\theta+ \cdots + n\cos n\theta$

Sum to $n$ terms and also to infinity of the following series:$$\cos \theta+ 2\cos 2\theta+ \cdots + n\cos n\theta$$the solution provided by the book is $$S_n=\frac{(n+1)\cos n\theta-n\cos(n+1)\theta-...
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1answer
75 views

Probably this :$\sum_{n=2}^{\infty}(-1)^n\frac{\arctan{(1-2^n)}\log n \tan{(1-2^{-n})}}{n^3\sqrt{n}\log \log n }$ is Euler constant

I'm always interesting to find some approach in the form of series or integral to get any known constant , In this once i have accrossed in my mind to use some trigonometrics functions in the form of ...
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4answers
77 views

Simplify $\sum_{k = 1}^n \tan(k) \tan(k - 1)$ by first proving $\tan(k)\tan(k - 1) = \frac{\tan(k) - \tan(k - 1)}{\tan(1)} - 1$

I have the following problem: Use the formula $$\tan(A - B) = \dfrac{\tan(A) - \tan(B)}{1 + \tan(A) \tan(B)}$$ to prove that $$\tan(k)\tan(k - 1) = \dfrac{\tan(k) - \tan(k - 1)}{\tan(...
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0answers
179 views

Sum of binomial coefficients in Gould tables

Consider the combinatorial identity by Gould, Table III, page 25, equation (6.13): $$\sum_{k=0}^{[\frac{n}{r}]}{n \choose rk}=\frac{2^n}{r}\sum_{j=1}^{r}\left(\cos{\frac{\pi j}{r}}\right)^n\cos{\frac{...
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39 views

Identical equation $\sin(\frac{nx}{2^{n-1}})=\sin x\times \sin(x+\pi/n)\times \cdot\cdot\cdot \times \sin(x+\frac{(n-1)\pi}{n}) $

Using $\;\;\sin(\frac{nx}{2^{n-1}})=\sin x\times \sin(x+\pi/n)\times \cdots \times \sin(x+\frac{(n-1)\pi}{n}) $ Simplify the following expression. $$\sum_{k=1}^{n}{\cot(x+\frac{(k-1)\pi}{n})}$$ ...
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1answer
39 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
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1answer
37 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+...
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1answer
111 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?
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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\...
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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.
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154 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 ...
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3answers
292 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 ...
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0answers
62 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} \...
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1answer
31 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 ...
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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})(...
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3answers
550 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 ...
2
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1answer
90 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. MY ATTEMPTION From LHS ...
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0answers
47 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 ...
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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{...
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1answer
45 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 ...
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1answer
70 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
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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} + \...
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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{...
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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
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1answer
50 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
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2answers
85 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
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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
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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 ...
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0answers
26 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
28 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
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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
65 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)...
2
votes
1answer
29 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
66 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
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2answers
45 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?