All Questions
Tagged with sequences-and-series trigonometry
820
questions
5
votes
3
answers
450
views
Evaluating $\frac1{m^2}\sum_{k=1}^m\sum_{l=1}^m\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)$ for natural $r\leq m$, and $A_k=\frac{2k-1}{2m}\pi$
IMO 1969 Longlist problem 38:
Let $r$ and $m$ ($r \leq m$) be natural numbers and $A_k = \frac{2k-1}{2m}\pi$.
Evaluate
$$\frac{1}{m^2}\sum_{k=1}^{m}\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k - rA_l)$...
- 356
1
vote
0
answers
35
views
What is the limit of this composed trig function (first question)? [duplicate]
This is my first question here so I don't know if I formatted this well. Please let me know how I could improve.
So, I was messing around on desmos, specifically with composing functions n times.
An ...
- 11
0
votes
1
answer
84
views
How to prove that $\sin\left(\frac{\pi}{2n}\right)\sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n}\pi\right)=1$? [closed]
How to prove that
$$\sin\left(\frac{\pi}{2n}\right)\sum_{k=1}^{n}\sin\left(\frac{2k-1}{2n}\pi\right)=1$$
?
- 305
3
votes
1
answer
94
views
Calculate $ \lim_{n \to \infty} \left( \sum_{k=1}^n \left( \sqrt{n^4 + k} \cdot \sin \left( 2\pi \cdot \frac{k}{n} \right) \right) \right) \ $
$$
\mbox{What is the value of this limit ?:}\quad
\lim_{n \to \infty}\sum_{k = 1}^{n}\sqrt{n^{4} + k\,}\
\sin\left(2\pi\,\frac{k}{n}\right)
$$
I tried looking for sum Riemann sums first, nothing.
I ...
- 1,311
0
votes
2
answers
222
views
summation of $\cos{x} + \cos{2x} + \cos{3x}$ and so on is $-\frac{1}{2}$
$$\cos{x} + \cos{2x} + \cos{3x} \ldots = y$$
$$ 2\cos{x} + 2\cos{2x} + 2\cos{3x} \ldots = 2y $$
By grouping every alternate term by $\cos{A} + \cos{B} = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2})$
$$ \...
- 11
2
votes
0
answers
50
views
Proving cluster points for a sequence [duplicate]
I have the sequence $a_n = ⌊\cos(\sqrt{n})⌋, n \in ℕ_0 $, and I have to determine the amount of cluster points. Of course, $\cos(x)$ oscillates between $-1$ and $1$, as $x$ varies, and therefore $⌊\...
- 51
2
votes
0
answers
40
views
Calculate $\sum\limits_{r=1}^\infty\frac{1}{16r^2-1}$ using the Fourier-Series of $|\sin(x)|$ [duplicate]
I have the following problem:
Calculate $\sum\limits_{r=1}^\infty\frac{1}{16r^2-1}$ using the Fourier-Series of $|\sin(x)|$
For this I already calculated the Fourier-Series of $|\sin(x)|$ to be
$$|\...
0
votes
1
answer
54
views
What am I doing wrong, when calculating sequence limit $\left|\sin \left(\pi \cdot \sqrt[3]{n^3+n^2}\right)\right|$
I need to calculate sequence limit of $\left\{a_n\right\}_{n=1}^{\infty}$ where $\left\{a_n\right\}_{n=1}^{\infty} = \left|\sin \left(\pi \cdot \sqrt[3]{n^3+n^2}\right)\right|$
$$
\begin{aligned}
&...
- 11
4
votes
1
answer
120
views
Proving that, for positive integers $k_i$, there exists $x_0\in[0,\pi]$, such that $\frac12+\sum_{i = 1}^m\cos(k_ix_0)<0$
$k_i$ is a positive integer, $i=1,\ldots,m$, please try to prove that there exist a point $x_0 \in [0,\pi]$, such that $\frac{1}{2}+\sum\limits_{i = 1}^m {\cos ({k_i}{x_0})} < 0$.
My attempt:
If $...
- 347
1
vote
1
answer
61
views
Does the series $\sum_{n=0}^\infty\arcsin(3^{-n})$ converge?
I have been working on this question for two days,
does $\sum_{n=0}^\infty\arcsin(3^{-n})$ converge?
I did that.
$$\arcsin x<\frac{x}{\sqrt{1-x^2}}$$ then i put $3^{-n}$ as $x$ , $3^{-n}\in (0,1]$
...
- 11
4
votes
3
answers
207
views
Finding the values of $x$ that satisfy $\sin x+\sin2x+\sin3x+\cdots+\sin nx\le\frac{\sqrt3}{2}$ for all $n$
If the exhaustive set of $x\in(0,2\pi)$ for which $\forall n$ the inequality $$\sin x+\sin2x+\sin3x+\cdots+\sin nx\le\frac{\sqrt3}{2}$$ is valid is $l_1\le x\le l_2$, find $l_1$ and $l_2$.
Let $\...
1
vote
1
answer
60
views
Proof that $\frac{d^a}{dx^a}\sin(x) = \sin(x+\pi a/4)$ iff these two infinite series are equivalent?
I am interested in analytically continuing the differentiation operation on functions, and I am currently focusing on trigonometric functions. Any high schooler can conjecture that $\frac{d^a}{dx^a}\...
- 453
0
votes
0
answers
39
views
$ \frac{\pi}{6} \int_{-\pi}^{\pi}e^{-jx\sin(\tau)}d\tau + \sum_{m=-\infty}^{+\infty}\frac{1}{2\pi m^2}\int_{-\pi}^{\pi}e^{2m\tau-x\sin(\tau)}d\tau $
Introduction
I'm grappling with an expression that intriguingly combines integrals and series, involving exponential functions with sinusoidal inputs. I'm curious about expressing this in terms of a ...
- 311
3
votes
0
answers
132
views
Calculate $\sum\limits_{k=1}^{\infty} \frac{\sin 2k}{3^k}$
I have solved it, but it does not match with the last part of solution. The logic is:
Let's consider complex series $\sum\limits_{k=1}^{\infty} \frac{\cos 2k + i\sin 2k}{3^k}$, imaginary part of which ...
- 105
5
votes
2
answers
81
views
Proof and example of the formulas for $\sin\left(\sum_{i=1}^\infty\theta_i\right)$ and $\cos\left(\sum_{i=1}^\infty\theta_i\right)$
Was reading the page on "List of trigonometric identities" on Wikipedia and came across an identity for sines and cosines of sums of infinitely many angles whose sum converges absolutely ...
- 356