For questions about or related to trigonometric series.

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2
votes
2answers
64 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
70 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
99 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
20 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
32 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
41 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
3answers
37 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
27 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
35 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
50 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
40 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
91 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 ...
6
votes
0answers
67 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 ...
0
votes
1answer
28 views

Determine the height of the tree

Dylan is using his clinometer to help him determine the height of a tree. He stands 6 m from the base of the tree and takes the measurement shown on the clinometer. Then, he measures the height of ...
-5
votes
1answer
41 views

Find $-3\vec u+4\vec v$ given $u=4\vec i + \vec j$ and $v=5 \vec i - 2\vec j$ [closed]

Find the vector $-3\vec u+4\vec v$ given vectors $$\vec u=4 \vec \imath + \vec \jmath$$ and $$v=5\vec \imath - \vec \jmath.$$ Write answer in the form $a \vec \imath + b \vec \jmath$.
0
votes
2answers
73 views

Could we solve $\int_{0}^{\infty}\sin(x)dx$ and what does it say about $\lim_{x\to\infty}\cos(x)$?

As the title states: Could we solve $\int_{0}^{\infty}\sin(x)dx$ and what does it say about $\lim_{x\to\infty}\cos(x)$? It is clear we can't solve this using the fundamental theorem of Calculus, but ...
3
votes
1answer
51 views

Example of an infinite sum of functions $f_n(x)$ that converges to $x$, is there a typo in my book?

I have a book that says the following: Let $f_1(x), f_2(x), \dotsc$ a sequence of bounded functions with $f_1(x) + f_2(x) + \dotsb = x$, for example $$ f_1(x) = \frac{\sin x}{x}, \; f_n(x) = ...
4
votes
0answers
148 views
4
votes
4answers
229 views

Sine of argument with large n approximation

I have worked an integral and reduced the integral to $$\frac{n \pi+\sin\left ( \frac{n \pi}{2} \right )-\sin\left ( \frac{3 \pi n}{2} \right )}{2n \pi}$$ I want to show that for $$n\rightarrow ...
2
votes
1answer
93 views

Summation relating factorial and cosine

How to simplify \begin{align*} \sum_{k=0}^{\infty}\left(-1\right)^{k}\frac{\left(2k\right)!}{4^{k}\left(k!\right)^{2}}\cos\left(kx\right) \end{align*} for $0\leq x <\pi$ ? I don't even know where ...
2
votes
1answer
73 views

How to find $\sum_{n=0}^\infty\sum_{k=0}^\infty\frac{(-1)^{n+k}}{(2k+1)(2n+1)^{2k+1}}$

I want to evaluate $$\sum_{n=0}^\infty\sum_{k=0}^\infty\frac{(-1)^{n+k}}{(2k+1)(2n+1)^{2k+1}}$$ From $$\arctan{x}=\sum_{k=0}^\infty\frac{(-1)^kx^{2k+1}}{2k+1}$$ I get ...
1
vote
3answers
56 views

Using Demoivre's Theorem prove that $ {\cos5 \theta} = 16{\cos^5 \theta} - 20{\cos^3 \theta} + 5{\cos \theta} $ .

$ {\cos5 \theta} = 16{\cos^5 \theta} - 20{\cos^3 \theta} + 5{\cos \theta} $ . Demoivre's Theorem $$ \{\cos \theta + i \sin \theta \}^n = \cos n\theta + i\sin n\theta $$ Where n is an integer . I ...
1
vote
2answers
63 views

Convergence or divergence of (i) $\sum_{n=1}^{\infty} \cos \frac 1n$,(ii)$\sum_{n=1}^{\infty} \cos \frac {1}{n^2}$

In case of (ii), As $n \rightarrow \infty$, $\cos \frac {1}{n^2} \rightarrow 1 \neq 0$. Hence this series diverges. In case of (i), $\cos \frac 1n = 1 - 2\sin^2 \frac {1}{2n}$. But since $\sum \sin ...
2
votes
2answers
60 views

Can you give a closed form or an asymptotic for $\sum_{m=0}^{k-1}\cos(\frac{2\pi m n}{k \log 2})$ for $k\to\infty$?

I want compute, in a closed form or an asymptotic (with a, big oh as, error term) this mean $$\delta_k(n):=\sum_{m=0}^{k-1}\cos(\frac{2\pi m n}{k \log 2})$$ defined for each integer $k\geq 1$. ...
0
votes
0answers
30 views

Sum of treble trigonometric Series

I am very much thankful to you if you can help me to find summation of following series. $$\sum_{i=1}^{N-2} \sum_{j=i+1}^{N-1} \sum_{k=j+1}^{N} [ \sin (ix-jx) + \sin (jx-kx) + \sin (kx-ix)]^2$$
0
votes
1answer
50 views

At what angle does the stone have to be hit? [duplicate]

What I have so far: 11 inches = 0.916667 feet Let a represent θ tan a = 0.916667 / 87 = 0.010536402 tan^-1(0.010536402) = 0.6036... rounded to 0.6 degrees. Is this correct?
0
votes
1answer
73 views

What is the value of x in this diagram?

So I'm pretty familiar with SOH-CAH-TOA but this question in particular looks a bit different and I'm not sure how to go about it. Thanks in advance!
0
votes
1answer
40 views

At what angle must one cut the board?

Given a 2" wide board and a 1 1/2" wide board, we would like to cut the narrower board at an angle $\theta$ so the cut is 2" long so the boards will fit together as shown in the diagram below. At what ...
1
vote
1answer
21 views

Finding a convergent majorant series

I have a series $$ \sum_{n=1}^\infty \left( \frac{1}{n^3} \cos(nt) - \frac{1}{(2n+1)^2} \sin(nt) \right) $$ and I have to find a majorant series to this series. The convergent majorant series I was ...
0
votes
0answers
12 views

Find convergent majorant series

I have a series $\sum_{n=1}^\infty \frac{1}{n^3} (\cos(nt) - 2 \sin(nt))$ and I want to find a convergent majorant series. I know that I have to find $k_n$ such that $|f_n(t)| = |\frac{1}{n^3} ...
0
votes
1answer
28 views

An inequality for $1+\cos(x-x_0)-\cos(\delta)$ in Trigonometric Series by Zygmund

On page 12 it's written that the function ($\delta>0$) $t(x)=1+\cos(x-x_0)-\cos(\delta)$ satisfies the following: $t(x)\ge 1$ in $I$ where $I = (x_0-\delta,x_0+\delta)$ $t(x)>1$ where $I'$ is ...
1
vote
1answer
24 views

Question on summation of the given series and applying limits.

Question is If the value of $\displaystyle \lim_{n\to\infty}\sum_{k=2}^{n}\cos^{-1}\left[\frac{1+\sqrt{(k-1)(k)(k+1)(k+2)}}{k(k+1)}\right]$ is equal to $\displaystyle\frac{120\pi}{m}$. Then find ...
0
votes
2answers
17 views

Is it possible to write a sinusoid of one frequency $f_1$ as a linear combination of sinusoids of different frequency than $f_1$?

If this is impossible (which I suspect it is), how can we formally show this using the notions of Fourier series and/or Fourier transform?
1
vote
2answers
40 views

Another way to express $\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$?

I believe that the sum $$\lim\limits_{m\to\infty}\sum_{n=1}^m\frac{\sin\left(2\pi n\left(1+\frac{1}{2m+2}\right)\right)}{n}$$ converges and it is about $1.85193$. Is there another way that this ...
3
votes
1answer
72 views

A weird property of $\sum_{k = 1}^{n} \sin k$

I was playing around with the sum $\sum_{k = 1}^{n} \sin k$, and using very loose rigour I arrived at the following: Proposition. Let $n \equiv n_0 \pmod {44}$ and $n_0 \equiv n_1 \pmod {6}$. Then ...
0
votes
1answer
12 views

Sum into closed form

I'm working with spectral approximations and I ran into this problem. Hope someone knows how to solve it! $(D_N)_{lj} = \frac{1}{N} \sum_{k=-N/2}^{N/2-1} i k e^{2 i k (l-j) \pi /N} $ ...
2
votes
3answers
62 views

Sum up trigonometric series [duplicate]

$$\cos \frac{2π}{2013} +\cos \frac{4π}{2013} +\cdots+\cos \frac{2010π}{2013} + \cos \frac{2012π}{2013}$$ How to sum it up? *Calculator is not allowed.
1
vote
2answers
30 views

What could the ratio of two sides of a triangle possibly have to do with exponential functions?

Name says it all. The two seem so unrelated? What's more, if I'm not mistaken the exponential version contains an imaginary part. I'm kind of ignorant about imaginary numbers, but does this mean that ...
4
votes
3answers
81 views

How to prove $\sum_{k=1}^n \cos(\frac{2 \pi k}{n}) = 0$ for any n>1? [duplicate]

I can show for any given value of n that the equation $$\sum_{k=1}^n \cos(\frac{2 \pi k}{n}) = 0$$ is true and I can see that geometrically it is true. However, I can not seem to prove it out ...
2
votes
1answer
66 views

Convergence of a sine series

Using Mathematica, we claim that the following series is convergent: $$\sum_{n=1}^{\infty}\frac{\sin(n^2 t)}{n}$$ Any idea how we prove this?
6
votes
0answers
85 views

On $\sum a^n \tan(n\theta)$

It is well known that $$\sum_{n=0}^{\infty} a^n \cos(n\theta) = \frac{1-a\cos(\theta)}{1-2a\cos(\theta)+a^2}$$ $$\sum_{n=0}^{\infty} a^n \sin(n\theta) = \frac{a\sin(\theta)}{1-2a\cos(\theta)+a^2}$$ ...
1
vote
1answer
48 views

prove that $\sum_{k=1}^{n}{\sin(kA)}$ [duplicate]

prove that $$\sum_{k=1}^{n}{\sin(kA)} = {{\cos({A\over2})-\cos(nA+{A\over 2})}\over 2\sin({A\over 2})}$$using Telescoping series. How do i go about doing this?
-4
votes
1answer
73 views

Infinite sum of cosine function [closed]

What does the following expression equal to $$\sum\limits_{n=1}^\infty \cos(n\cdot\theta)=\text{?}$$
1
vote
1answer
31 views

Find the limit of $\lim_{j\rightarrow \infty}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$

Can anyone give a hint on how to see if the following has a limit? $f$ stands for frequency. $$\lim_{j\rightarrow \infty}\sin^2(2^{j-1}\pi f)\prod_{i=0}^{j-2} \cos^2(2^i\pi f)$$ I've tried a few ...
3
votes
1answer
36 views

The sum $\sum_{n\leq x}\sum_{\substack{1\leq k\leq n \\ gdc(k,n)=1}}cos^2\pi \frac{k}{n}$ diverges as $x$, when $x$ tends to infitity

I want to know if it is possible find an easy proof (this is without an use of an strong result) of Question. Prove that the following sum diverges as $x\to\infty$ $$\sum_{n\leq ...
0
votes
0answers
33 views

$\sum_{n=0}^x \sin(n)$?

I saw an article somewhere asking about the question $\sin 1 + \sin 2 + \sin 3 + ... +\sin 90$, and the answer the article gave was ~$57.76..$ But a quick graph in Desmos reveals that the titular sum ...
0
votes
0answers
74 views

Express ∑sin(kx)/k as the sum of a function and the integral of another function

I wish to find functions $g_n$ and $h_n$ defined on $(0,π)$such that 1. $\displaystyle\sum_{k=1}^n \frac{\sin(kx)}{k}= g_n(x)-\int_x^π h_n(t) \, dt$ 2. $\displaystyle h_n(x)≥0$ For all $x \in ...
0
votes
1answer
62 views

$\sum_{n=0}^{\infty} \sin (nx) = \cot(x/2)$?

I noticed this trend playing around in desmos, that the series $\sin x + \sin 2x + \sin 3x +...$ tends to $\cot(\frac{x}{2})$, is this identity correct? Here is an example:
0
votes
1answer
32 views

Prove convergence of $\int^{\infty}_{0}\ t^{z-1}cos(t)dt$ and $\int^{\infty}_{0}\ t^{z-1}sin(t)dt$

For a complex analysis problem set I am trying to show that the integrals $$\int^{\infty}_{0}\ t^{z-1}cos(t)dt \quad and \quad \int^{\infty}_{0}\ t^{z-1}sin(t)dt $$ is convergent for ...
0
votes
1answer
61 views

Find $n$ such that $\tan 1$ and its Taylor series up to $n$ agree to 1000 decimal places

I know the Taylor series for $\tan x$ is, $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ I am trying to find a value for $n$ such that $|\tan 1 ...