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Questions tagged [trigonometric-series]

For questions about or related to trigonometric series.

0
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2answers
38 views

Evaluate the sum of the infinite series $1+\cos x + \cos^2 x + \cos ^3 x …$ for $0<x<\pi$

Evaluate the sum of the infinite series $1+\cos x + \cos^2 x + \cos ^3 x ...$ for $0<x<\pi$ So am I correct in thinking that $$1+\cos x + \cos^2 x + \cos ^3 x ...=\sum ^\infty _{n=0} \cos^n x$$ ...
1
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2answers
33 views

Prove the following infinite trig product by induction.

Prove by induction, that for a positive integrer $n$, that $$\cos x \times \cos2x \times \cos 4x \times \cos 8x ... \times\cos (2^nx) = \frac{\sin(2^{n+1}x)}{2^{n+1}\sin x}$$ So to start I'm gonna ...
0
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4answers
49 views

Is $\sum\limits_{n=1}^\infty \arctan \frac x {n^2}$ continuous?

Let $$ f(x) = \sum\limits_{n=1}^\infty \arctan \frac x {n^2} $$ I need to check whether $f : \mathbb R\to \mathbb R$ is continuous. Of course, if it converges, $f(x) = -f(-x)$, so I will be only ...
2
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2answers
68 views

Finding all real $x$ such that $1+\sum_{j=1}^n\sin{\frac{j\pi x}{n+1}} = 0$, where $n=18$.

The task is to find all $ x \in \mathbb R $ such that $$ 1 + \sum_{j=1}^n \sin{\frac{j\pi x}{n + 1}} = 0, \qquad n = 18 $$ What I have tried Using the following formulas: $$1. \sin{x} + \sin{y} ...
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0answers
24 views

References for a proof of a Jackson's inequality?

Let $g:[0,2\pi]\to \mathbb{C}$ which is $\mathcal{C}^k([0,2\pi],\mathbb{R})$ and periodic. If $\mid f^{(k)}(x) \mid\le 1$ then for each $n\in \mathbb{N}^*$, there exists a trigonometric polynomial $T_{...
6
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2answers
156 views

$\sum_{n=1}^\infty a_n \cos nx$ unbounded near $0$ if $\sum a_n$ diverges?

If $a_n$ is a decreasing positive sequence and tends to $0$, and given$$\sum_{n=1}^\infty a_n=+\infty$$ can we prove that $$\lim_{x\rightarrow 0}\sum_{n=1}^{\infty} a_n \cos nx =+\infty$$ or at least ...
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0answers
50 views

Infinite Sum of a Converging Series of Inverse Cosines

I was wondering if anyone knew a way to find the value of d for the following converging infinite sum: $$ \sum_{n=2 }^{\infty}\arccos \left ( -\frac{d^{n+1}+d^{2n}+d^n-d}{(d^{n-1}-1)(d^n-d)} \right ) ...
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0answers
26 views

What type of waveform is described by this fourier series

A signal with a time of 0.2 miliseconds is described by the below equation: $$ f(x) = \frac{1}{2} + \sin(\omega_0x) - \frac{1}{2}\sin(2\omega_0x) +\frac{1}{3}\sin(3\omega_0x) - \frac{1}{4}\sin(4\...
2
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1answer
56 views

Is $f(x) \leq x$ for $0 \leq x \leq \pi$ when sine series $f(x)$ are used to approximate $x$ based on derivatives at $x=0$?

This is a simpler "cousin" question to Would sine trigonometric series $f(x)$ for approximating $g(x) = x$ always be $f(x) \leq x$ for $0 \leq x \leq \pi$? . I am asking this as a separate question, ...
1
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0answers
40 views

Would sine trigonometric series $f(x)$ for approximating $g(x) = x$ always be $f(x) \leq x$ for $0 \leq x \leq \pi$?

I am trying to use set of $c_k$ with $k \in \mathbb{N}$ such that $f(x) = \sum_{k=1}^{M} c_k \sin kx \approx x$. $c_k$ is determined by setting $$f^{(2k+1)}(0) =0,\,\, k=1,..,n$$ $$f'(0) =1$$ $$f^{(...
3
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2answers
82 views

Prove $\sum_{k=1}^m \cot^2 k\pi/(2m+1)=m(2m-1)/3$

Prove that $$ \sum_{k=1}^m \cot^2 \frac{k\pi}{2m+1}=\frac{m(2m-1)}{3} $$ I have tried to use $$\sin\left((2m+1)x\right)= \left(\sin^{2m+1}x\right) \cdot \left(\sum_{j=0}^m (-1)^j \binom{2m+1}{2j+1}\...
1
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3answers
89 views

Non zero solution of $3x\cos(x) + (-3 + x^2)\sin(x)=0$

How can I find exact non-zero solution of $3x\cos(x) + (-3 + x^2)\sin(x)=0$. Simple analysis and the below plot show that the equation has an infinite number of non-zero solutions.
1
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3answers
56 views

Series expansion (likely Maclaurin) of integral

As someone who is trained formally in physics, and not mathematics, I have become rusty in series expansions of special integrals (and/or) identities that exist regarding integrals of inverse ...
4
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3answers
126 views

Compute $\sum\limits_{j = 0}^{m - 1} \left(c_j + 1\right)\ln\left(c_j + 1\right)$ where $c_j = \cos\left(\frac{\pi}{2m}\left(1 + 2j\right) \right)$

As part of solving: \begin{equation} I_m = \int_0^1 \ln\left(1 + x^{2m}\right)\:dx. \end{equation} where $m \in \mathbb{N}$. I found an unresolved component that I'm unsure how to start: \begin{...
2
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1answer
37 views

Fallacious moving of powers resulting with a correct trigonometric series identity.

Prove that $$ \\ \sum_{r=0}^n \left( \frac{ (-1)^r {n \choose r} } {n+r+1} ( \sin^{2(n+r+1)}x + \cos^{2(n+r+1)}x )\right) = \sum_{r=0}^n \frac{ (-1)^r {n \choose r} } {n+r+1} $$ for all values of ...
1
vote
1answer
47 views

Summation of $\arccos\left(\frac{n^2+r^2+r}{\sqrt{(n^2+r^2+r)^2+n^2}}\right)$

I found this question in a book, and cannot solve it. I have to find the the sum $$S_n=\sum_{r=0}^{n-1} \arccos\left(\frac{n^2+r^2+r}{\sqrt{(n^2+r^2+r)^2+n^2}}\right)$$ I tried converting this to $\...
4
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4answers
71 views

How to check if the sum of infinite series is convergent?

I have this exercise where I need to find if the sum of infinite series is convergent: $\sum_{n=1}^ \infty \frac{(\sin^2(x) - \sin (x) +1)^n}{\ln(1+n)} $ for x $ \in (\pi/2,\pi) $ Now I decided to ...
2
votes
2answers
50 views

Calculation of Complex Trigonometric Summation

Evaluation of $$\sum^{n}_{k=1}\frac{\tan(x/2^k)}{2^{k-1}\cdot \cos(x/2^{k-1})}.$$ Try:Let $$S=\sum^{n}_{k=1}\frac{\sin(x/2^k)}{2^{k-1}\cos(x/2^{k-1})\cdot \cos(x/2^k)}$$ $$S=\sum^{n}_{k=1}\frac{\sin\...
0
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0answers
47 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 ...
1
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0answers
123 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^{...
3
votes
3answers
100 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 ...
1
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4answers
583 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 $4\sqrt{3}$....
0
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1answer
43 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}{...
0
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1answer
19 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 ...
2
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1answer
107 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\...
2
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1answer
61 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. ...
1
vote
1answer
118 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. $$$$ ...
0
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1answer
73 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 ...
0
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2answers
167 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
31 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 $-...
2
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2answers
65 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 ...
1
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0answers
47 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 ...
0
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1answer
35 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)\...
0
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1answer
43 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!}\...
8
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2answers
184 views

$\lim\limits_{s\to0^+}\sum_n\frac{\cos\left(\pi\frac{n}{m}\right)}{n^s}$ & $\lim\limits_{s\to0^+}\sum_n\frac{\sin\left(\pi\frac{n}{m}\right)}{n^s}$

$(1).$ Show that: $$ \lim_{s\to0^+}\,\left[\sum_{n=1}^{\infty}\cos\left(\pi\frac{n}{m}\right)\frac{1}{n^s}\right]=\color{red}{-\frac{1}{2}} \quad\colon\space\forall\,m\in\mathbb{N}^{+}\tag{1} $$ ...
0
votes
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
votes
3answers
130 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....
1
vote
0answers
29 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
votes
1answer
98 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
votes
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{-...
1
vote
1answer
60 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 ...
2
votes
2answers
149 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
votes
1answer
54 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}{...
3
votes
2answers
141 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 ...
6
votes
2answers
123 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) $ ...
4
votes
2answers
220 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-...
0
votes
1answer
82 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 ...
2
votes
4answers
84 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(...
7
votes
0answers
192 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{...
0
votes
0answers
47 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. $$\sum_{k=1}^{n}{\cot(x+\frac{(k-1)\pi}{n})}$$ ...