For questions about or related to trigonometric series.

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10
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0answers
140 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
votes
2answers
40 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
48 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
votes
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
votes
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
vote
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
votes
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 ...
3
votes
3answers
87 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 ...
0
votes
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
54 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
votes
4answers
59 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
votes
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
vote
0answers
37 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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
0answers
32 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
vote
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
vote
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
votes
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
votes
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
113 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
vote
0answers
32 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
vote
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
votes
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
68 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 ...
4
votes
0answers
73 views

find the closed form of $\sum_{k=1}^\infty\left(\frac{\sec{kz}}{k^2}\right)^2$

How to evaluate $\displaystyle\sum_{k=1}^\infty\left(\frac{\sec{(k\pi\sqrt{5})}}{k^2}\right)^2$? In general, how to find the closed form of infinite series ...