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 $\sum_n c_n e^{inx}$.
859
questions
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Prove that , $\sin \theta + \cos \theta > 1$ [closed]
Prove that $\sin \theta + \cos \theta > 1$.
You can prove it with geometrical terms and others.
4
votes
1
answer
101
views
Proving a trigonometric finite sum $\sum_{k=1}^N(-1)^k(\cos \frac{k\pi}{N})^{N-m}(\sin\frac{k\pi}{N})^m=(-1)^{m/2}\frac{N}{2^{N-1}}$
How to prove this following formula?
$\sum_{k=1}^N(-1)^k(\cos \frac{k\pi}{N})^{N-m}(\sin\frac{k\pi}{N})^m=(-1)^{m/2}\frac{N}{2^{N-1}}$
for m is even, and $0$ for m is odd.
If we know $\sum_{k=1}^N(-1)^...
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1
vote
0
answers
46
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Hint for summation is needed [duplicate]
$$
\newcommand{\arccot}{\operatorname{arccot}}
\arccot(2)+\arccot(8)+\arccot(18)+\arccot(32)+\ldots=?
$$
this the question.
First find a relation of $2,8,18,32,\ldots$
$$
a_1=2, a_2=8, a_3=18, ...
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0
votes
0
answers
55
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What does Euler mean when he says "Let the arc z be infinitely small ; there will be sin.z=z and cos.z=1..."?
I am referring to, specifically the sin.z=z and cos.z=1. I am reading Euler's Introductio In Analysin Infinitorium, Vol. I, Ch. VIII and he jumps from binomials to ...
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2
votes
0
answers
105
views
A trigonometric sum [closed]
For $k=0,\cdots, m$ and $l=0,\cdots 2m+1$ let us put
$$ \alpha_{kl}=\frac{4k-4l+1}{4(m+1)}\pi\quad \beta_{kl}=\frac{4k+4l+3}{4(m+1)}\pi $$
and
$$x_{kl}=\frac{1}{4}\Big(\frac{1}{\sin\alpha_{kl}}+\frac{...
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6
votes
1
answer
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If $\sum_{i=1} ^{2022}\sin^{-1}(x_i) = 1011\pi$, then find $\sum_{i=1}^{2022} x_i$
A question is given in my book which I'm unable to solve.
If $\displaystyle\sum_{i=1}^{2022} \sin^{-1}(x_i) = 1011\pi
$, then what is the value of $\displaystyle\sum_{i=1}^{2022} x_i$?
Answer of the ...
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0
votes
0
answers
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Are good kernels trigonometric polynomials?
We know :
Definition: A kernel $K_n$ is 'good' if they are integrable and satisfy the following conditions:
$\int_{-\pi}^{\pi}K_n(x)dx=1$
$\int_{-\pi}^{\pi}|K_n(x)|dx\le A$ for some $A>0$
For ...
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0
votes
0
answers
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Can I use an identity to go from $\sin(2\pi(\omega(t + h) + \beta))\sin(2\pi(\omega t + \beta))$ to $\frac{1}{2}\cos(2\pi \omega h)$?
In my time series homework I had to compute the periodogram of a periodic series and in the solutions we have a step that goes from $\sum_{t=1}^{n-h} \sin(2\pi(\omega(t + h) + \beta))\sin(2\pi(\omega ...
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1
vote
2
answers
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Counting roots of oscillatory functions
Consider the functions $f_m : [0,1] \to \mathbb{R}$ defined by
$f_m(x) = \displaystyle\sum_{n=1}^{m} \frac{\sin(nx)}{1+\sin^2(n)}$
for every natural number $m$. My task is to find the number of ...
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2
votes
0
answers
72
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$\int_0^\pi t^2\sin(t)\sin(n(\sin(t)-t\cos(t)))dt$ for alternate Goat problem Fourier sine series solution
The goat problem has the following transcendental equation:
$$\sin(a)-a\cos(a)=\frac\pi2,a=1.905695729\dots$$
If $f^{-1}(x)$ is odd, its Fourier sine series of period $\left[-\frac L2,\frac L2\right]$ ...
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1
vote
0
answers
31
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Proving some trigonometric series are convergent
I have to prove that if $(a_k)_k$ is decreasing and tending to 0, then, for all $\epsilon>0$, the series $\sum a_k \sin(2\pi kx), \sum a_k \cos(2\pi kx)$ and $\sum a_k e^{2\pi ikx}$ are all ...
2
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2
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142
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How to prove identity $\sum_{k=0}^{n-1}\tan^2\left(\theta+{k\pi\over n}\right)=n^2\cot^2\left({n\pi\over2}+n\theta\right)+n(n-1)$?
Looking at Jolley, Summation of Series, formula 445:
$\sum_{k=0}^{n-1}\tan^2\left(\theta+{k\pi\over n}\right)=n^2\cot^2\left({n\pi\over2}+n\theta\right)+n(n-1)$
How can one prove this?
Considering $\...
- 1,267
0
votes
0
answers
45
views
Cumulative sum of trigonometric functions
I have two known time series $ Y, 𝜭 $
$Y = \left\{ Y_i \right\}_N \hspace {0.2 in} Y_i > 0 \hspace {0.2 in} -1 \leq Y_{i+1} - Y_i \leq 1 $ for all $i$
$𝜭 = \left\{ 𝜭_i \right\}_N $
Let ...
- 51
5
votes
1
answer
210
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Is this a correct method for finding Fourier series coefficients?
I'm trying to find Fourier series coefficients $c_n$ for given signal
$x(t)=\cos(2\pi t)+\cos(4\pi t)$
Solution:
$$x(t)=\sum_{n=-\infty}^\infty c_n e^{j \frac{2\pi}{T_0}nt}$$
$$\cos(2\pi t)+\cos(4\pi ...
0
votes
1
answer
65
views
Any idea on how WolframAlpha did this sum?
So I've been trying to work out this sum for quite a while:
WolframAlpha unfortunately won't supply step by step proofs for this for some reason...
As for how i tried to prove this:
I looked at sin((...
1
vote
1
answer
42
views
Find minimum n :$ \prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{\sqrt{27k^3+54k^2+36k+8}}\right)}{\arctan \left( \frac{1}{\sqrt{3k+1}} \right)}>2000$ [closed]
We've got to find the minimum value of n for which
$$ \prod_{k=1}^n \frac{\arcsin\left( \frac{9k+2}{\sqrt{27k^3+54k^2+36k+8}}\right)}{\arctan \left( \frac{1}{\sqrt{3k+1}} \right)}>2000$$
So I just ...
- 131
10
votes
1
answer
229
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Is there an identity for $\sum_{k=0}^{n-1}\tan^4\left({k\pi\over n}\right)$?
Is there a simple relation for $\sum_{k=0}^{n-1}\tan^4\left({k\pi\over n}\right)$
like there is for $\sum_{k=0}^{n-1}\tan^2\left({k\pi\over n}\right)$?
Looking at Jolley, Summation of Series, formula ...
- 1,267
1
vote
1
answer
54
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Prove the inequality $\sum_{n=1}^\infty \left(\int_E \cos 2nxdx\right)^2\le \pi |E|$.
I am trying to prove this inequality but I am not sure where does the coefficient $\pi$ come from.
Desired Inequality: $$\sum_{n=1}^\infty \left(\int_E \cos 2nxdx\right)^2\le \pi |E|,$$ where $E$ is a ...
- 141
1
vote
0
answers
72
views
Prove an inequality involving trigonometric series
I am reading a proof, in which there is an inequality that I cannot prove.
Let $\{\phi_n\}$ be a sequence of reals. Let $N$ be an integer $\ge 2$.
Let $p(t)=\sum_{n=0}^N b_n \cos(nt+\phi_n)$.
Desired ...
- 141
0
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0
answers
46
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Trigonometric Curve Fitting with Phase Shift
I have a similar situation as described her
Initial Guess for Trigonometric Curve Fitting
However, now I would like to fit measured data as described in the linked post with a polynoimal of the form
$$...
- 133
1
vote
1
answer
54
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Initial Guess for Trigonometric Curve Fitting
I have from measurements data which should behave like a cosine function, but doesn't because of some physical effects.
For a cosine function with maximum $y_{\text{max}}$ and minimum $y_{\text{min}}$ ...
- 133
1
vote
0
answers
59
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Sum of inverse tangents of the roots of a polynomial
Let p(x) = $$n_0x^{2k+1}+n_0x^{2k}+n_1x^{2k-1}+n_1x^{2k-2}+n_2x^{2k-3}+n_2x^{2k-4}+.........+n_{k+1}x+n_{k+1}$$
where k is a positive integer and the coefficients of every two adjacent terms in the ...
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votes
1
answer
97
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Infinite sum of sine
I do not know exactly whether this summation $\displaystyle \sum_{n=1}^{\infty} |\sin(x_{n})|$, with $x_{n}$ approaches $0$, cannot be $\infty$ or not.
But I guess it is actually divergent but does ...
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1
vote
2
answers
48
views
limit of $1- \cos(ka)$
I am reading a text which says
for $ka \ll 1$:
$1 - \cos(ka) \approx \frac{1}{2}(ka)^{2}$ but I fail to understand why this is so.
Could someone shed light on this?
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1
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3
answers
140
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Maclaurin series of $\sin(5x^2)$
I was requested to find the Maclaurin series of $\sin(5x^2)$. I attempted to find the derivatives of this function in hopes of finding a pattern. However, the derivatives become more and more ...
- 2,646
1
vote
0
answers
53
views
Cosine Function As Infinite Product
I'm trying to expand cosine function by using the Mittag-Leffler theorem which is introduced in Arfken 7ed.:
$$f(z)=f(0)\exp\left(\frac{zf'(0)}{f(0)}\right)\prod_{n=1}^{\infty}\left(1-\frac{z}{z_n}e^{...
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5
votes
2
answers
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Integrating from $\int_0^\infty \frac{\sin(x)}x dx$ with year $10$ mathematics?
I am currently a year $10$ student self studying a level further maths. One day, my friend sent me this funny photo, from the photo we can easily know that glass beer = $10$, burger = $5$, and cup ...
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1
vote
0
answers
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sums of imaginary exponentials
For two exponentials, we have:
$$e^{i\alpha} + e^{i\beta} = 2\exp\left(i\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right), \tag{1a}$$
$$e^{i\alpha} - e^{i\beta} = 2i\exp\left(i\frac{...
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0
votes
1
answer
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views
Series of tan inverse x for x > 1
The Maclaurin series of the Tan inverse of $x$ works well for $-1 \leq x \leq 1$ but it breaks down afterwards because the curve takes a different.
What function defines the curve after $x = 1$? ...
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0
votes
0
answers
107
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Evaluating the trace of two discrete Sine and Cosine transforms
We recall that discrete Cosine transform of type 5 and discrete Sine transform of type 8 are given as follows:
$$C_5=\left(\cos kl\frac{\pi}{n-\frac{1}{2}}\right)_{0\leq k,l\leq n-1}~,~S_8=\left(\sin (...
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0
votes
1
answer
58
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Asymptotic average of cosine function
I am wondering whether the following equality is true and how it can be proved:
$$
\lim_{T \to \infty} \frac{1}{T} \sum_{t=1}^T \cos(2t)=0.
$$
- 359
1
vote
2
answers
60
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Infinite sum of Spherical Bessel function and Cosine at odd index
I've recently encountered the expression \begin{equation}
\sum_{n = 0}^\infty J_{2n+1}(a)\cos[(2n+1)b].
\end{equation}
I'm famililar with the even form of this expression, which as a closed form ...
- 11
2
votes
1
answer
163
views
Infinite Sum of an Inverse Trig Expression
I am attempting to find either a closed form for the following infinite sum, or failing that, the value $p$ for which the sum converges to $2\pi$ (somewhere around $0.82$?).
$$\sum_{i=1}^\infty \...
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2
votes
1
answer
140
views
Is there any subsequence of the sequence $(\frac{\cos(\alpha - n \beta) - \lambda \cos(\alpha + n \beta)}{ \cos^n(\beta) })$ that converges?
Let's take $\alpha$ and $\beta$ two reals in $(0, \frac{\pi}{2})$.
Let's take $\lambda \in (0,1)$.
Let's define the sequence $(u_n)$ as follows:
\begin{eqnarray}
u_n & = & \frac{\cos(\alpha - ...
- 51
6
votes
1
answer
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Series expansion of $x$ in terms of $\sin(\frac{x}{n})$ [closed]
I am looking at expanding $y = x$ for $ 0 < x < \pi$ in terms of sine functions of the type $\sin(\frac{x}{n})$ where $n \in \mathbb N$. This looks a lot like the Fourier series but I could not ...
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0
votes
1
answer
576
views
Find the $Z$-transform of $\sin (\alpha k), k \ge0$
Find the $Z$-transform of $\sin (\alpha k), k \ge0$
Solution
Could anyone please explain how we got the second step (in terms of $e$ and $i$) after writing it in basic $Z$-transform notation? And how ...
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2
votes
1
answer
102
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$f(x)=\sum_{k=0}^{n}\binom{n}{k}\sin^{k}x\sec^{k-n}{x}.$
Let $n$ be a non-negative integer and let $$f(x)=\sum_{k=0}^{n}\dbinom{n}{k}\sin^{k}x\sec^{k-n}{x}.$$
Prove that $f(x)$ is periodic and find its amplitude.
I don't really know how to start, and all I ...
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0
votes
1
answer
473
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On the Fourier series expansion of $\sin(\pi x)$ periodic on $(-\frac{1}{2}, -\frac{1}{2})$
For an Undergrad. sophomore Math Methods class I am taking this session, we have recently covered Fourier series expansions. I made good progress in one of the exercises, but I was stuck for days on ...
- 145
0
votes
0
answers
26
views
Trigonometric polynomails dense in $C(T)$ with period $1$ instead of $2\pi$
It is well known, that trigonometric polynomials of the form $$P(t) = \sum^{N}_{n=-N}c_ne^{int}$$ are dense in $C(T)$, which is the set of all continuous, complex, $2\pi$ periodic functions.
Is it ...
0
votes
1
answer
35
views
Closed form expression for discrete-time sum of a cosinusoid
I'm facing a problem in digital signal processing and am wondering if there is a closed form expression for the sum
$$Y[n] = \sum_{k=0}^{n-1}\cos(\frac{2\pi k}{f}),$$
where $n$ < $f$.
In case I ...
- 3
0
votes
0
answers
66
views
Central Limit Theorem for irrational rotation
If $ X=\mathbb{R}/\mathbb{Z}$ be the unit circle with $\mu$ Lebesgue
measure, and suppose $\alpha\in X$ irrational, and $T:X\rightarrow X$
is defined by $T(x)=x+\alpha$. we want to find a function $f\...
- 1
1
vote
0
answers
32
views
A complicated sum of cosines involving sums inside the cosine arguments
Let $n$ be an integer greater than $1$. How do I show that
$$\sum_{L_{1}=0}^{n-2}\sum_{L_{2}=0}^{L_{1}}\left(-1\right)^{L_{1}+L_{2}}\left(1+\left(-1\right)^{L_{1}+L_{2}}\cos\left(\pi\sum_{k=L_{2}+1}^{...
- 73
1
vote
1
answer
99
views
Need advice on approximating the sum of a trigonomitric series which (I think) has no analytical solution
I've encountered a maths problem in a programming project I'm working on. I've tried a lot of things already, and I'm feeling very swamped with maths that is way above my head.
I need to find:
$$f_N(x)...
- 21
0
votes
0
answers
71
views
$ \sum_{k=1}^{n} \arctan(k)=? $ Using argument of complex number
Question:
\begin{align} \sum_{k=1}^{n} \arctan(k)=? \end{align}
My approach:
\begin{align} \prod_{k=1}^{n} (1+ik)=a+ib\qquad(1)\end{align}
\begin{align} \Rightarrow arg \Bigg(\prod_{k=1}^{n} (1+ik)\...
user1027837
14
votes
3
answers
273
views
Proving $\left|\sin x\right|\leq\frac{(2m+1)!}{2^{4m}(m!)^2}\left[\binom{2m}{m} - \sum_{k=1}^{m} \frac{2}{4k^2-1} \binom{2m}{m+k} \cos(2kx) \right]$
Update. Based on @Exodd's answer, it turns out that the upper bound is equal to
$$ T_{2m}(\cos x) = \sum_{k=0}^{m} (-1)^k \binom{1/2}{k}\cos^{2k} x, $$
where $T_{2m}(x)$ is the degree $2m$ Taylor ...
- 150k
0
votes
0
answers
24
views
Notation for Expresion of Tangent of Summation of Angles
I am proving, assuming this expression leads to something correct, that the tangent of a summation of angles is an expression involving sums of products of the tangents of each angle, like this:
$$
\...
- 2,945
1
vote
0
answers
45
views
Bound on amount of minima of finite Fourier sum
Consider a finite Fourier sum of the form $$f(\theta) = \sum_{i=1}^n r_i \cos(m_i \theta) \,,$$ where $n \geq 1$ is an integer, the $r_i$ are positive real numbers and the $m_i$ are integers. Is there ...
- 188
0
votes
1
answer
19
views
Showing that $R= \frac{1}{n}\sum_{k=1}^n\cos(\theta_k - \overline{\theta})$ given a system of equations
Let angles $\theta_1,\dots,\theta_n$ be given and define $C = \frac{1}{n}\sum_{k=1}^n\cos(\theta_k)$ and $S = \frac{1}{n}\sum_{k=1}^n\sin(\theta_k)$. Then, there exists a value $\overline{\theta}$ ...
- 445
0
votes
0
answers
27
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Relating complex logarithm power series to the arctan power series
I've been trying to directly convert the expected complex logarithm power series to the arctan series without success. I expected some interplay between the terms so that imaginary terms would cancel, ...
- 91
0
votes
0
answers
43
views
Is it possible to express $\sin(\frac{k\pi}{4})$ as something of the form $(-1)^{f(k)}$?
For example $\cos(k\pi)=(-1)^k$ similarily, $\sin(\frac{k\pi}{2})=(-1)^\frac{k-1}{2}$ when k is odd, and 0 otherwise. Is there a similar representation for $\sin(\frac{k\pi}{4})$
