For questions about or related to trigonometric series.

learn more… | top users | synonyms

0
votes
0answers
12 views
3
votes
1answer
43 views

$\lim\limits_{i\mapsto \infty} \sum_{i=1}^\infty \sin(\theta_{i+1}-\theta_{i})$ = ? , where $\theta_{i}= \pi\sum_{j=0}^i \frac{1}{{(2)}^j}$

$\lim\limits_{i\mapsto \infty} \sum_{i=1}^\infty \sin(\theta_{i+1}-\theta_{i})$ = ? $\theta_{i}= \pi\sum_{j=0}^i \frac{1}{{(2)}^j} = \pi \left(2 - \frac{1}{2^{i}}\right)$ The limit ...
-2
votes
1answer
34 views

a Proof onInverse Trigonometry

Ff $\arcsin x + \arcsin z + \arcsin z = 1.5\pi$, Prove that $x^{2006}+y^{2007}+z^{2008}-\frac 9{x^{2006}+y^{2007}+z^{2008}}=0$.
1
vote
1answer
22 views

How does $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ become $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$?

How can you rewrite $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ to $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$? What rules have been used? I need it on ...
0
votes
2answers
78 views

Find the number of solutions of $\sin x+2 \sin 2x- \sin 3x=3$

In $(0 \:\:\pi)$Find the number of solutions of $$\sin x+2 \sin 2x- \sin 3x=3$$ The equation can be written as $$\sin x+4 \sin x \cos x=3+\sin 3x$$ i.e. $$\sin x(1+4\cos x)=3+\sin 3x$$ i.e., ...
1
vote
1answer
41 views

Trigonometric series convergence

I was interested in evaluating $$ \sum_{n=1}^{\infty} \frac{\cos n}{n+k} $$ I saw with computation that many of such series converge. Is there a general result? I've tried using Taylor expansion of ...
16
votes
0answers
309 views

Simplify the sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$? - a sum shows all primes $\le N^2$

I was looking for a closed form but it seemed too difficult. Now I'm seeking help to simplify this sum. The 50 bounty points or more will be awarded for any meaningful simplification of this sum. I ...
0
votes
2answers
38 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 ...
0
votes
2answers
59 views

Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right) $$ ...
0
votes
0answers
22 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. $$ ...
3
votes
4answers
98 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 ...
2
votes
2answers
45 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
54 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
81 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
56 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
41 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
31 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
votes
1answer
32 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
57 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
30 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
55 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
61 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
66 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
29 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
46 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
51 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
37 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
22 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
102 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
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
140 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
34 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
88 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
40 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
vote
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
vote
0answers
37 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 ...