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Questions tagged [trigonometric-series]

For questions about or related to trigonometric series.

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32 views

math of Diffusion ; diffusion through membrane

A liquid diffuses through a porous membrane of thickness L. If the concentration c(x,t) is maintained at c1 on the x=0 side of the membrane and c2 on the x=L side of the membrane, determine c(x,t) on ...
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0answers
31 views

Fourier series of Heaviside step function?

Let us say we have the Heaviside unit step function $\Theta(t-t^\prime)$. I want to calculate its Fourier series $$ \Theta(t-t^\prime)=\frac{1}{T}\sum_{n,m}\Theta_{\omega_n,\omega_m}e^{-i\omega_n t}e^{...
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3answers
92 views

Proof formula for $(\sin(x))^n$

I currently try to proof the following equation: $$(\sin(x))^n=\sum_{k=0}^{n}{a_k\cos(kx)+b_k\sin(kx)}$$ with $a_0,...,a_k$ and $b_0,...b_k$ being real numbers for each $n$. I tried to proof this ...
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4answers
275 views

Prove that $\frac{1}{\sin\frac{\pi}{15}}+\frac{1}{\sin\frac{2\pi}{15}}-\frac{1}{\sin\frac{4\pi}{15}}+\frac{1}{\sin\frac{8\pi}{15}}=4\sqrt{3}$

I'm trying to calculate the expression: $$\frac{1}{\sin\frac{\pi}{15}}+\frac{1}{\sin\frac{2\pi}{15}}-\frac{1}{\sin\frac{4\pi}{15}}+\frac{1}{\sin\frac{8\pi}{15}}$$ and show that it is equal $4\sqrt{3}$....
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1answer
36 views

Taylor series expansion for $\cos(2x)$ about $\frac{\pi}{8}$

So, knowing that $$f(x) = \sum_{n=0}^\infty \frac{f^n(a)(x-a)^n}{n!}$$ For my case I write $$\cos(2x) = \sum_{n=0}^\infty \frac{\frac{d^n(cos(\frac{\pi}{4}))}{d(\frac{\pi}{8})^n}(x-\frac{\pi}{8})^n}{...
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1answer
16 views

Asymptotic approximation/expansion for arccosine function?

Trying to find a 3-term asymptotic expansion for $z=cos^{-1}(x)$, as $x\rightarrow1^-$. Found a lot of examples online for inverse tangent, cosine, etc. but have yet to find any guidance on the ...
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1answer
106 views

Are the partial sums for $\sum_{n=1}^{\infty}\sin(n^a)$ bounded for $a\geq1$ and unbounded for $0<a<1$?

I know that the partial sums of $$\sum_{n=1}^{\infty}\sin(n)$$ are bounded between $\frac{\cos\left(\frac{1}{2}\right)-1}{2\sin\left(\frac{1}{2} \right)}$ and $\frac{1+\cos\left(\frac{1}{2} \right)}{2\...
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1answer
50 views

Bounding $\left|\sum_{\nu=n+1}^{\infty}a_\nu cos(\nu x)\right|$

I'm reading Zygmund's Trigonometric Series (precisely Lemma 6.6 of chapter 12), and I'm struggling to understand the following detail. $(a_n)$ here is a sequence of positive numbers and decreasing. ...
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1answer
70 views

Find the Sum of the Series Using Complex Exponentials

Find the sum of the series $\sum_{n=0}^{\infty}\frac{\cos(nx)}{2^{n}}$ and $\sum_{n=0}^{\infty}\frac{\sin(nx)}{2^{n}}$. Hint: Rewrite the trigonometric functions using complex exponentials. $$$$ ...
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1answer
69 views

Proof of the formula for $\sin \theta + 2\sin 2\theta +\cdots + n\sin n\theta$

I'm looking to show that $$\sin \theta + 2\sin 2\theta +\cdots + n\sin n\theta = \frac14\left(\;(n+1)\sin (n\theta) - n \sin((n+1)\theta)\;\right)\csc^2\left(\frac{\theta}{2}\right)$$ So far, I ...
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2answers
93 views

The integer part of $\sum_{k=0}^{44}\frac{1}{\cos(k^\circ)\cos((k+1)^\circ)}$ [duplicate]

What is the integer part of the number $$\sum_{k=0}^{44}\frac{1}{\cos (k^\circ)\cos((k+1)^\circ)}$$ I tried to solve it using partial fractions but could not get a result. Please help me out.
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2answers
28 views

$1 + \cos2C - \cos2A - \cos2B=4\sin A\sin B\sin C$.

How can I prove this equation? $1 + \cos2C - \cos2A - \cos2B=4\sin A\sin B\sin C$ if we know that $A$, $B$, $C$ are a triangle's angles. I have come to the point where on the left side I have $-...
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2answers
58 views

Using Leibniz on $\sum_{n=1}^\infty \sin(\pi \sqrt{n^2+1})$

Using Leibniz on $\sum_{n=1}^\infty \sin(\pi \sqrt{n^2+1})$ So the question actually is how to rewrite $\sin(\pi\sqrt{n^2+1})$ in the form of $(-1)^n\times a_n$ so that I can apply Leibniz and decide ...
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0answers
39 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
33 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
38 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
45 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
21 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: $\...
2
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3answers
128 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
27 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 $\min(f_N)=?$ ...
3
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1answer
86 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 ...
3
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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
58 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
145 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
52 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
119 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
122 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
214 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
78 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
81 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
184 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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45 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\...
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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
117 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
70 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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0answers
158 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
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3answers
295 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
65 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
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1answer
32 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
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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})(...
13
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3answers
556 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
votes
1answer
95 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 ...
2
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0answers
48 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
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1answer
35 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
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1answer
51 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
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1answer
71 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
49 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
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2answers
81 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{...