For questions about or related to trigonometric series.

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0
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0answers
18 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ ...
0
votes
2answers
31 views

Product of trigonometric polynomials is a trigonometric polynomial

A trigonometric polynomial was defined as $$f(x) = \frac{a_0}{2} + \sum_{k=1}^{n}(a_k cos(kx) + b_k sin(kx))$$ I heard somewhere that trigonometric polynomials have a ring structure, i.e. a product ...
3
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4answers
90 views

How to evaluate this limit using Taylor expansions?

I am trying to evaluate this limit: $\lim_{x \to 0} \dfrac{(\sin x -x)(\cos x -1)}{x(e^x-1)^4}$ I know that I need to use Taylor expansions for $\sin x -x$, $\cos x -1$ and $e^x-1$. I also realise ...
14
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0answers
206 views
+50

Find a closed form formula for $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes.

I meant by "closed form formula" a formulate that doesn't have summation or has very few terms. Maybe there's a better term for this meaning. I found this function that has very interesting property ...
2
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2answers
41 views

Evaluate the following trignometric sum

I am interested in the following sum $$\sum_{\text{even } n=-\infty}^{\infty}\left(-\cos^2x\delta_{n,0}+\cos x\left(\frac{1-\cos x}{\sin x}\right)^{|n|}\right).$$ Wolfram alpha returns answer ...
1
vote
2answers
51 views

Why do I get different results for the same integral?

The variables $ a, b, s, c $ are constants, so: $$ \int \left ( a \cos(s + cx) - b \sin(s + cx) \right ) dx = \frac{a\sin(s + cx) + b\cos(s + cx)}{c} +C $$ But if $c=0$ then: $$ \int \left ( a ...
4
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0answers
78 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
0
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0answers
22 views

On a certain series of cosines

For natural numbers $p$ and $q$, compute the value of $$\displaystyle \sum_{k=0}^{q-1} \cos^{p} \left(\dfrac{2\pi k}{q}\right).$$ I got the answer $$\dfrac{q}{2^p} \sum_{l=1}^{p} \binom{p}{l} ...
2
votes
2answers
53 views

prove $\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$

Today I found the identity : $$\sum_{k=1}^n(-1)^{k+1}\cos^{2n+1}\left(\dfrac{k\pi}{2n+1}\right)=\frac{1}{2}-\frac{2n+1}{2^{2n+1}}$$. How to prove or disprove this? Thank you.
1
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1answer
38 views

Computation of an inverse trigonometric series using complex numbers

The following is a popular question (in competitive exams) in India: Compute the value of $S=\displaystyle \sum_{k=1}^{\infty} \tan^{-1}\left( \dfrac1{2k^2}\right)$. I can compute the value by ...
2
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1answer
29 views

How to get from $\sum_{n=0}^N (a_n \cos{nx} + b_n \sin{nx})$ to $\sum_{-N}^{N} c_n e^{inx}$?

I'm currently reading Baby Rudin, and I'm in the section of Chapter 8 that covers Fourier series. There is one line that I just can't figure out for the life of me, and I can't find anything online ...
0
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1answer
30 views

How to appraoch solving this series?

I am given the following series and asked to solve it. $\sum\limits_{n=1}^\infty \dfrac{(-2)^n}{(2n+1)!}$ I recognize that this series is somewhat similar to the Taylor series for $sinx$ which is ...
-1
votes
1answer
48 views

Prove $a_n = \sin(n\pi /2)$ does not converge to $0$

Not sure how to prove it doesn't converge to $0$. Prove by contradiction? How do I assume that the sequence converges to $0$ since if I do the partial sums it does end up adding to $0$.
0
votes
1answer
24 views

How to find the scaling with N for FWHM and MAX of $F(X) = (1+\cos((2N+1)\pi X)/(1+\cos(\pi X))$?

Given a function $F(X) = (1+\cos((2N+1)\pi X)/(1+\cos(\pi X))$. Peaks are at odd $X$ integer values. How to find the scaling with $N$ of the FWHM (full width half max) and peak max? For example for ...
1
vote
3answers
54 views

How to find $\sum_{k=1}^nk\sin\left(k\pi/n\right)$ [duplicate]

I would appreciate if somebody could help me with the following problem Q: How to find ? $$\sin\left( \frac{\pi}{n}\right)+2\sin\left( \frac{2\pi}{n}\right)+\cdots+ n\sin\left( ...
2
votes
1answer
114 views

new $\arctan$ series working for any $x$?

Using Log series, we can write: $$ \frac{1}{2 i} \left( \text{Log} \left( 1 + e^{\text{i2t}} \right) - \text{Log} \left( 1 + e^{\text{-i2t}} \right) \right) = \frac {1} {2 i}\left ( \sum _ {k = ...
1
vote
2answers
55 views

Uniform convergence of a trigonometric series

Let $L>0$ be a constant. With what coefficients $\alpha_k$ and $\beta_k$ does the trigonometric series $$ \alpha_0 +\sum_{k=1}^{\infty} \left[\alpha_k \cos\left( \frac{k\pi x}{L} \right) ...
2
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4answers
61 views

How to transform the equation $\sum_{k=1}^{n}{sin(kx)}$? [duplicate]

How to prove this equation or how to transform the left part to the right: $$\sin(x)+\sin(2x)+\dots +\sin(nx)=\frac{\sin(\frac{nx}{2})\sin\left(\frac{(n+1)x}{2}\right)}{\sin\left(\frac{x}{2}\right)}$$ ...
0
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0answers
25 views

For a real number $r>0$ for the function $\cot(z)$ has a Laurent series where 0<|z|<r. What is the largest possible r?

Okay so I know that $\cot(z)$ expands as: $$ \cot(z) = \frac{1}{z} - \frac{z}{3} - \frac{z^3}{45} - \ldots$$ But I'm unsure how to find the largest $r$. I'm assuming this is the point where it ...
1
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0answers
39 views

Laurent series for $\cot(\pi z)/z^2$

I'm looking for the series for $\frac{cot(\pi z)}{z^2}$ using the residue theory, where the function denotes a circle about the origin with a radius of $k+\frac{1}{2}$ I found that the residues of ...
1
vote
2answers
67 views

Quick Method to Calculate the Maclaurin Series of $\frac{1}{\sqrt{\cos{x}}} $

I am supposed to calculate the maclaurin series for $\frac{1}{\sqrt{\cos{x}}} $ but I can't seem to figure out an efficient way to go about doing this.
4
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0answers
38 views

Sum of Squares of Tangents [duplicate]

Find $S$ if $S=\tan^21^\circ+\tan^23^\circ+\tan^25^\circ+\dots+\tan^289^\circ$. I tried converting all tangents above $45^\circ$ to cotangents and added them with tangents with the same angle, ...
1
vote
1answer
46 views

Error of Taylor Series?

Part of my assignment is to find the third degree Taylor Series of $\tan(x)$ about $\pi/4$ and then estimate the error of this approximation when evaluated at 0.75. Finding the series was easy ...
0
votes
1answer
36 views

Find $\sum_{k=0}^6cot({\pi\over 21} + {k\pi\over 7})$

Find $$\sum_{k=0}^6cot({\pi\over 21} + {k\pi\over 7})$$ I don't know how to do this at all. I initially started using complex numbers but couldn't get it. Please give a simple solution.Thanks.
0
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0answers
25 views

Defining trigonometric functions for obtuse angle

I realized that following is apparently true: we had definitions of sine, cosine, etc. for angles of right triangle. Then, one suddenly draws X and Y axes, incorporates negative numbers, and sees that ...
0
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1answer
21 views

How to show: $-\sum_{k=-\infty}^\infty \frac{i*k*(-1)^k}{1+k^2}e^{-i*k*x}=2\sum_{k=1}^\infty \frac{k(-1)^{k-1}}{1+k^2}sin(kx)$

I am struggling to show below in a big question: $-\sum_{k=-\infty}^\infty \frac{i*k*(-1)^k}{1+k^2}e^{-i*k*x}=2\sum_{k=1}^\infty \frac{k(-1)^{k-1}}{1+k^2}sin(kx)$ Tried to analyse with geometric ...
0
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1answer
31 views

Converting certain complex exponentials to trigonometric functions

The original question is: Show that $$f(x)=\sum_{k=-\infty}^\infty c_k e^{-i kx}=\frac{2\sinh(\pi)}{\pi} \sum_{k=1}^\infty \frac{(-1)^{k-1}k}{1+k^2}\sin(kx)$$ where $\displaystyle ...
6
votes
1answer
96 views

Simplify Product of sines

Is there a way simplify this product? $$ \sin\left({n} \frac{\pi}{2}\right) \sin\left({n} \frac{\pi}{3}\right) \sin\left({n} \frac{\pi}{4}\right) ...\sin\left({n} \frac{\pi}{n-1}\right) $$ And, is ...
1
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0answers
80 views

Value of $\tan(1^{\circ})+\tan(2^{\circ})+\tan(3^{\circ})+…+\tan(89^{\circ})$

Is there any method to find value of $$\tan(1^{\circ})+\tan(2^{\circ})+\tan(3^{\circ})+\dots+\tan(89^{\circ})$$ without using calculator. To find the same sum for sine and cosine , I used De-Movier's ...
3
votes
1answer
24 views

$\sum_{n=2}^\infty \frac{e^{2\pi in t}}{n \log n}$

Does any know how to show that $\sum_{n=2}^\infty \frac{e^{2\pi in t}}{n \log n}$ is in $L^{2}(\mathbb T)$
6
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0answers
33 views

Certain symmetrized product of cosines - can it be transformed into more manageable form

I am interested in the following expression: $$ F_{k_1,\ldots,k_n}(t):=\sum_{\sigma\in S_n}\cos(\sigma(1)k_1t)\cos(\sigma(2)k_2t)\cdots\cos(\sigma(n)k_nt) $$ where $k_1, \ldots, k_n$ are natural ...
0
votes
1answer
26 views

Trigonometric 0001 sequence

Need help in generating particular sequence (0001000100010001...) using trigonometric functions formula (e.a. "cos"). For example, the formula for sequence 0101010101... is: $f(x)=\cos(\pi x/2).$ For ...
7
votes
1answer
138 views

Why does $\sum_{n=0}^{\infty} \cos^n(n)$ converges?

Consider the series $$\sum_{n=0}^{\infty}\cos^n(n)$$ I think that the root test is inconclusive, because $$\limsup_n \sqrt[n]{|\cos^n(n)|}=\limsup_n|\cos(n)|\leq 1$$ once we can approximate $\pi$ ...
1
vote
2answers
32 views

Simplifying Trig Product in terms of a single expression and $n$

I participated in a math competition at a nearby university yesterday, and there was one ciphering problem that no one correctly simplified. As the title suggests, we were asked to simplify the ...
1
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2answers
86 views

Coincidence that series of arctan is alternating series of artanh?

I noticed that the power series for $\arctan$ is the alternating series of that for $\operatorname{arctanh}$. Does it have a special meaning or even some kind of special importance?
1
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1answer
26 views

Sum of weighted cosine functions (2)

I had written a question but I did not formulate the problem correctly (sorry). The last question is in this link. I am writing the problem again and I would appreciate any help. Assume $\theta_{ij}$ ...
0
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2answers
28 views

Sum of weighted cosine functions

I have got a question and I would appreciate if one could help! I want to maximize a function that after some algebraic manipulation results in the sum of weighted cosine with different phases. ...
0
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1answer
39 views

Vaughan's identity, a didactic example

I know that Vaughan's identity is one of the methods used in analityc number theory, I would like see an example, I say a simple example of application of this theorem for encourage to study the ...
2
votes
2answers
77 views

$|\sin(\sin( \cdots \sin(x)\cdots))|$ ($N$ times) is always $\leq|\sin( \cdots \sin(1)\cdots)|$ ($N-1$ times)

Is this inequality always true? $$ \bigl\lvert\,\underbrace{\sin(\sin(\cdots \sin}_{N\text{ times}}(x)\cdots))\bigr\rvert\le\bigl\lvert\,\underbrace{\sin(\sin( \cdots \sin}_{N-1\text{ ...
5
votes
1answer
73 views

Find the limit of this sequence

Suppose $$ f_n(x)=\sum_{k=1}^n \frac{\cos(kx)}{k}, $$ and let $a_n=\min_{x \in [0,\pi/2]} f_n(x)$, find $\lim_{n \to\infty} a_n$. I wrote a program and found that the $\arg\min_{x \in [0,\pi/2]} ...
4
votes
4answers
114 views

Evaluation of $\,\displaystyle \lim_{n\rightarrow \infty}\sum_{k=1}^n\sin \left(\frac{n}{n^2+k^2}\right)$

Evaluation of $\displaystyle \lim_{n\rightarrow \infty}\sin \left(\frac{n}{n^2+1}\right)+\sin \left(\frac{n}{n^2+2^2}\right)+\cdots+\sin \left(\frac{n}{n^2+n^2}\right)$ $\bf{My Try::}$ We can ...
1
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0answers
33 views

Integral of sums of basic trigonometric polynomials

My textbook (RCA, Rudin) asserts (in the proof of theorem 5.15) that $$ \lim_{n\to\infty}\lVert{\sum_{k=-n}^n}e^{ikt}\rVert_1=\infty. $$ Why is this true? I tried using Euler's formula to reduce the ...
1
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0answers
36 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
0
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0answers
43 views

How can I change a summation of cosines to a product of cosines for higher degree functions?

I was wondering if I can get an alternate form of a sum over cosines $$\sum_{n=1}^m\cos{f(n)}$$ and I found that I can. We must however make a modification to the upper limit with $m\rightarrow2^m$. ...
1
vote
2answers
39 views

Show that the sequence $\left(\frac{x^{2k+1}}{(2k+1)!}\right)$ where $x\in(0,1)$ decreases monotonically and converges to $0$

I have to show that sequences $\left(\frac{x^{2k+1}}{(2k+1)!}\right)$ where $x\in(0,1)$ and $\left(-\frac{x^{2k+1}}{(2k+1)!}\right)$ where $x\in(-1,0)$ decrease monotonically and converge to $0$. I ...
0
votes
0answers
29 views

Expand a $\arctan(x)$ function [duplicate]

I want to expand a function $\arctan(x)$ as a polynomial form. I know that I can use Taylor expansion in the case of x <1. But in my case, the x can be pretty large. Is there any way to expand or ...
1
vote
1answer
36 views

Page 72 of Courant and Hilbert's Methods of Mathematical Physics, Vol 1.

We have the following identities: $$ \beta_\nu = b_\nu -\frac{1}{2}(b_{\nu-1}+b_{\nu+1}),\ \ \ \ (\nu=2,3,4,\ldots)\\ \beta_1=b_1-1/2 b_2 $$ $$s_n(x)=\sum_{\nu=1}^n b_\nu \sin(\nu x) \\ ...
2
votes
2answers
70 views

MacLaurin series for $9\sec(3x)$

A question I've been given asks me to find the first 3 non-zero terms of the MacLaurin series for the function: $y = 9sec(3x)$ Looking at old questions on this forum, I think that this is supposed to ...
1
vote
1answer
53 views

How do you find the sum: $\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$

How do you find the sum: $$\sum_{r=1}^6 \tan^2\left(\frac{r \pi}{n}\right)$$ I managed to solve this question using complex numbers so I thought I'd share the solution. If you know of any better ...
7
votes
2answers
107 views

How do you find the value of $\sum_{r=0}^{44} \tan^2(2r+1)$?

Problem: Find the value of $$\sum_{r=0}^{44} \tan^2(2r+1)$$ Note: The angles here are in degrees. I don't know how to solve this question because trigonometric simplifications didn't get me ...