2
votes
1answer
38 views

Sum of fractions of squared sines

I'm trying to prove the following approximate identity for $p$ integer: $$ \sum_{l=1}^m\frac{\sin^2\left(\frac{\pi l}{p}\right)}{\sin^2\left(\frac{\pi l}{mp}\right)}\sim \frac{m^2(p-1)}{2}+O(m) $$ ...
-1
votes
0answers
18 views

solving a complex fourier series

while solving for co efficient of a complex fourier series i came across this form cos(0.5*n*pi)+i*sin(0.5*n*pi) is there any way to simplify this? Note: N is an integer
0
votes
1answer
47 views

Why do different trig functions sum differently?

Why does the $\sum_{n=1}^{\infty} \sin (\frac 1 {n^2})$ converge but the $\sum_{n=1}^{\infty} \cos (\frac 1 {n^2})$ diverge?
3
votes
2answers
143 views

Does $\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $?

Stuck on homework problem (not this), if I can prove as a lemma that the sequence $$\sin(\sin(\sin\cdots(\sin1)\cdots) \rightarrow 0 $$ then I'm done. It's monotonic and decreasing and bounded by 0 ...
0
votes
1answer
47 views

Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
3
votes
2answers
149 views

Proof of $\sin^2(x) + \cos^2(x)=1$ using series

I have to prove the following identity $\sin^2 (x) + \cos^2(x)=1$. I can easily prove this, but this exercise is given in the section introducing the series expansions for $\sin(x)$ and $\cos(x)$ and ...
5
votes
2answers
103 views

Concerning the sequence $\Big(\dfrac {\tan n}{n}\Big) $

Is the sequence $\Big(\dfrac {\tan n}{n}\Big) $ convergent ? If not convergent , is it properly divergent i.e. tends to either $+\infty$ or $-\infty$ ? ( Owing to $\tan (n+1)= \dfrac {\tan n + \tan ...
7
votes
2answers
96 views

Showing that $\sum_{k=0}^{\infty} a^{k} \cos(kx) = \frac{1- a \cos x}{1-2a \cos x + a^{2}}$ without using complex variables

I was challenged a couple of weeks ago by Ron Gordon to show that $$\sum_{k=0}^{\infty} a^{k} \cos(kx) = \frac{1- a \cos x}{1-2a \cos x + a^{2}} \ \ (|a| <1)$$ without using complex variables. ...
1
vote
2answers
76 views

Trigonometric Series Proof

I am posed with the following question: Prove that for even powers of $\sin$: $$ \int_0^{\pi/2} \sin^{2n}(x) dx = \dfrac{1 \cdot 3 \cdot 5\cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots 2n} \times ...
1
vote
3answers
38 views

How to prove that $\Sigma_{n=1}^{N-1}{\sin^2\frac{p\pi{}n}{N}} = \frac{N}{2}$

Here $p$ is a known integer constant ($p > 0$). I know that this is true for a fact (checked numerically in Matlab and it holds), but I'm just not able to prove it. I noticed a similar problem ...
1
vote
2answers
136 views

How to prove $(\space|\sin n|\space)$ does not converge?

How do we prove that $(\space|\sin n|\space)$ is not convergent ? There is a beautiful proof of the non-convergence of $(\sin n)$ by considering the identities $\sin (n+1)=\cos1\sin n+ \sin 1 \cos n ...
1
vote
1answer
74 views

What are the coefficients of these trigonometric sums?

I have two functions that I'm working on. The first is: $$ \begin{align} \cos x &= (\cos 1)^3 \cos(3-x) \\ &{}+ 3 (\cos 1)^2 (\sin 1) \sin(3-x) \\ &{}- 3 (\cos 1) (\sin 1)^2 \cos(3-x) \\ ...
0
votes
1answer
78 views

Convergence of a sequence in trigonometric functions

Is the sequence $(a_n)$ defined as $$a_n :=\dfrac {\sin\Big( \dfrac {\pi}4+\dfrac n2 \Big)\sin\Big(\dfrac{n+1}2 \Big)}{\sin\Big(2n+2\Big)\sin\Big( \dfrac {\pi}4+\dfrac {n-1}2 \Big)\sin\Big(\dfrac{n}2 ...
3
votes
1answer
68 views

Find $\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$

Find the following limit: $$\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\cos k+\sum_{k=1}^{n}\sin k}{\prod_{k=1}^{n}\cos k\sin k}$$ The numerator can be simplified by using Euler's formula and the sum of ...
3
votes
7answers
293 views

Prove $ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $

$$ \cos\left(\frac{2\pi}{5}\right) + \cos\left(\frac{4\pi}{5}\right) + \cos\left(\frac{6\pi}{5}\right)+ \cos\left(\frac{8\pi}{5}\right) = -1 $$ Wolframalpha shows that it is a correct identity, ...
4
votes
1answer
106 views

Convergence of $\sum\frac{\tan(nz)}{n^2}$ to an analytic function…what if $z\in \mathbb{R}$?

For which values of $z$ does $$\sum_{n=1}^\infty \frac{\tan(nz)}{n^2}$$ converge? For which values of $z$ is the limiting function analytic? One can show, as in this answer, that ...
2
votes
2answers
79 views

Show these approximations of $\cos$, $\sin$ and $\tan$ are exact.

A while back I was looking for an approximation to $\cos(x)$ and I constructed a polynomial with zeros in the same places as the first few zeros of $cos(x)$: $$c_n(x) = \frac{\prod_{i=1}^n ...
0
votes
1answer
59 views

Sum with sine function

Can anybody help me in solving the following sum $$\sum\limits_{n=1}^\infty \frac{\sin(b\log n)}{b}$$I tried using the expansion of sine function but got stuck there.
4
votes
2answers
127 views

How does the Herglotz trick work?

There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions ...
19
votes
1answer
235 views

How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?

How can I prove the following identity? $$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$$
3
votes
0answers
144 views

Partial fraction development of $\cot \pi z$

"Compute the values $\sum_{n=1}^\infty \dfrac{1}{n^2}$ and $\sum_{n=1}^\infty \dfrac{1}{n^4}$ by comparison to the partial fraction development of $\cot \pi z$." I'm not sure what "the partial ...
0
votes
1answer
86 views

Why tan(1/z) has a non-isolated singularity at z=0?

Can someone please explain me this concept. Any sort of help will be highly appreciated.
3
votes
1answer
164 views

Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...
-2
votes
2answers
73 views

Is the sum of this series convergent? and how to find the sum?

I don't know if this is a duplicate, but I can't seem to find how to prove that the sum of this series is convergent, this series is actually the area hyperbolic tangent function. An additional ...
0
votes
2answers
196 views

Convergent subsequences of $x_n = \sin n$ and $y_n = \cos n$…

As in title, $x_n = \sin n$ and $y_n = \cos n$. Can we find some index sequence $\{n_k\}$ such that both $\{x_{nk}\}$ and $\{y_{nk}\}$ converge? (Not necessarily to the same limit) I'm fairly ...
1
vote
1answer
76 views

Summation of trigonometric functions

So consider a summation of ai cos (x + phi_i) where i ranges from 1 to N. Could we describe this summation as a single cosine function? Or the sum of two cosine or sine functions? How would we do this ...
0
votes
1answer
37 views

Showing a certain series equals a certain trig function in complex analysis

I was reading a a complex analysis proof that was showing that for a fixed $\alpha$ s.t. $Im(\alpha)>0$ we have $\sum \limits_{m=-\infty}^{\infty} \frac{1}{(\alpha +n)^2}=(-4\pi^2)\sum ...
3
votes
0answers
63 views

Proof of a series containing normal as well as hyperbolic functions

Show that $$\sum _{n=0}^{\infty}\frac{\sinh(\pi n \sqrt 2)- \sin(\pi n \sqrt 2)}{n^3{\cosh(\pi n \sqrt 2})- \cos( \pi n \sqrt 2)}=\frac{ \pi^3}{18 \sqrt 2}$$ I have no hint as to how to even start.
7
votes
1answer
86 views

Prove $\frac {1}{\cos 0^\circ \cdot \cos 1^\circ} + \ldots +\frac {1}{\cos 88^\circ \cdot \cos 89^\circ}= \frac{\cos 1^\circ}{\sin 1^\circ}$

Prove the following identity: $$\frac {1}{\cos 0^{\circ} \cdot \cos 1^{\circ}} + \ldots +\frac {1}{\cos 88^{\circ} \cdot \cos 89^{\circ}} = \frac{\cos 1^{\circ}}{\sin 1^{\circ}}$$ After hours of ...
0
votes
1answer
209 views

Is sine of angles greater than 90 degrees a convention?

The sine function is defined as the opposite side of the angle in question over the hypotenuse of the $90^\circ$ triangle. $$\sin(â) = \frac{opposite}{hypotenuse} \tag{$0^\circ<â<90^\circ$}$$ ...
2
votes
1answer
113 views

show that $ \sum \frac{1}{n}e^{-n^2 t}\sin{(nx)}$ converges in $L^2([0,\pi])$ to $ \sum \frac{1}{n}\sin{(nx)}$

I would like to show that $$ \sum_{\substack{n=1 \\ n \text{ odd}}}^\infty \frac{1}{n}e^{-n^2 t}\sin{(nx)}$$ converges in $L^2([0,\pi])$ to $$ \sum_{\substack{n=1 \\ n \text{ odd}}}^\infty ...
3
votes
1answer
139 views

How does a taylor series of a binomial function equals a trigonometric function? [closed]

Any proof or derivation for the sinx and cosx function would be help. Image taken from http://en.wikipedia.org/wiki/Taylor_series
3
votes
2answers
215 views

Showing $|\sum_{k=1}^n\frac{\sin(kx)}k|<2\sqrt{\pi}$

For any real $x$ and positive integer $n$, is it true that: $$\left|\sum_{k=1}^n\frac{\sin(kx)}k\right|<2\sqrt{\pi}\quad ?$$ Please justify.
4
votes
2answers
351 views

Compute $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$

Compute the series $\sum_{j=1}^k\cos^n(j\pi/k)\sin(nj\pi/k)$ Hint: the answer is in fact 0
30
votes
1answer
599 views

Why does this ratio of sums of square roots equal $1+\sqrt2+\sqrt{4+2\sqrt2}=\cot\frac\pi{16}$ for any natural number $n$?

Why is the following function $f(n)$ constant for any natural number $n$? $$f(n)=\frac{\sum_{k=1}^{n^2+2n}\sqrt{\sqrt{2n+2}+{\sqrt{n+1+\sqrt ...
6
votes
1answer
191 views

Find the sum : $\frac{1}{\cos0^\circ\cos1^\circ}+\frac{1}{\cos1^\circ \cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+…+$

Find the sum of the following : (i) $$\frac{1}{\cos0^\circ \cos1^\circ}+\frac{1}{\cos1^\circ\cos2^\circ} +\frac{1}{\cos2^\circ \cos3^\circ}+......+\frac{1}{\cos88^\circ \cos89^\circ}$$ I tried : ...
-3
votes
1answer
93 views

How to simplify this equation?

How to simplify this equation? I know that: $\sin^2 p+\cos^2 p = 1$ But how to go further?
5
votes
2answers
106 views

Sequence and Series - If $a_n =\int^{\frac{\pi}{2}}_0 \frac{\sin^2nx}{\sin^2x}dx,$…

If $\displaystyle a_n =\int^{\frac{\pi}{2}}_0 \frac{\sin^2nx}{\sin^2x}dx, $ then find the value of $$\begin{vmatrix} a_1 & a_{51} & a_{101} \\ a_2 &a_{52} & a_{102}\\ a_3 & ...
10
votes
2answers
351 views

Find the sum : $\sin^{-1}\frac{1}{\sqrt{2}}+\sin^{-1}\frac{\sqrt{2}-1}{\sqrt{6}}+\sin^{-1}\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}+\cdots$

Problem : Find the sum of : $$\sin^{-1}\frac{1}{\sqrt{2}}+\sin^{-1}\frac{\sqrt{2}-1}{\sqrt{6}}+\sin^{-1}\frac{\sqrt{3}-\sqrt{2}}{\sqrt{12}}+\cdots$$ My approach : Here the $n$'th term is given ...
1
vote
2answers
57 views

Finding the relation between function x,y,z - trigo problem

Problem : For $\displaystyle 0 < \theta < \frac{\pi}{2}$ if $$\begin{align}x &= \sum^{\infty}_{n =0} \cos^{2n}\theta \\ y &= \sum^{\infty}_{n =0} \sin^{2n}\theta\\ z &= ...
2
votes
3answers
82 views

Need to find function related to Knoedel numbers that satisfies these conditions

I need to find the continuous function $f(x)$ that satisfies $f(0)=0$ and: $$\frac{f(\sin(\pi/6))^2}{\sin^4(\pi/6)}=135$$ $$\frac{f(\sin(\pi/4))^2}{\sin^4(\pi/4)}=63$$ ...
1
vote
3answers
113 views

Does $a_n = \cos\left(n\ln \left(1+\frac{\pi}{n}\right)\right)$ converge?

I want to check if a sequence converges or diverges. The sequence is the following: $$a_n = \cos\left(n\ln \left(1+\frac{\pi}{n}\right)\right)$$ I though of maybe using sandwich theorem, but can I ...
4
votes
1answer
98 views

Prove a Trigonometric Series

Question: $$\cot^{2}\frac{\pi }{2m+1}+\cot^{2}\frac{2\pi }{2m+1}+\cdots+\cot^{2}\frac{m\pi }{2m+1}=\frac{m(2m-1)}{3}$$ $m$ is a positive integer. Attempt: I started by showing that ...
4
votes
1answer
120 views

Finite Series - reciprocals of sines

Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be ...
3
votes
1answer
37 views

Is $S=\sum_{r=1}^\infty \tan^{-1}\frac{2r}{2+r^2+r^4}$ finite?

Problem: If $$S=\sum_{r=1}^\infty \tan^{-1}\left(\frac{2r}{2+r^2+r^4}\right)$$ Then find S ?? Solution: I know that $\tan^{-1} x + \tan^{-1} y= \tan^{-1} \frac {x +y} {1-xy} $ But I have no idea ...
3
votes
3answers
434 views

Trigonometric identities using $\sin x$ and $\cos x$ definition as infinite series

Can someone show the way to proof that $$\cos(x+y) = \cos x\cdot\cos y - \sin x\cdot\sin y$$ and $$\cos^2x+\sin^2 x = 1$$ using the definition of $\sin x$ and $\cos x$ with infinite series. thanks...
15
votes
1answer
449 views

How to prove $\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\frac{\pi-1}{2}$

One of my classmates challenged me to solve $\displaystyle\sum\limits_{n=1}^{\infty}\frac{\sin n}n=\;?$ With a simple c program I found that $\displaystyle\sum\limits_{n=1}^{1048576}\frac{\sin ...
29
votes
4answers
532 views

$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$

Is it possible to calculate the following infinite sum in a closed form? If yes, please point me to the right direction. $$\sum_{n=1}^\infty(n\ \text{arccot}\ n-1)$$
3
votes
1answer
103 views

Proof the following trig series

Prove that $$\frac{ \sin x}{ \cos x}+\frac{\sin2x}{\cos^{2}x}+\frac{\sin3x}{\cos^{3}x}+\cdots+\frac{\sin nx}{\cos^{n}x}=\cot x-\frac{\cos(n+1)x}{\sin x \cos^{n}x}$$ I am not necessarily looking for a ...
11
votes
1answer
186 views

Need help with calculating this sum: $\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$

I need help with calculating this sum: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$