For questions about or related to trigonometric series.

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6
votes
1answer
720 views

Sum of tangent functions where arguments are in specific arithmetic series

By looking through an book, I found this interesting series To prove that: $$\tan(\theta)+\tan \left(\theta+ \frac{\pi}{n} \right) + \tan(\theta + \frac{2\pi}{n}) + \dots + \tan \left (\theta + ...
5
votes
1answer
162 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 ...
2
votes
4answers
186 views

Number of iterations to reach cosine's fixed point

I was messing around with my calculator the other day when I saw something interesting happen. Whenever I repetitively took the cosine of any number, it always ended up on a particular number ...
11
votes
2answers
234 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: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
3
votes
2answers
250 views

Taylor Series of $\tan x$

I found a nice general formula for the Taylor series of $\tan x$: $$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n - 1} $$ where $B_n$ are the Bernoulli ...
6
votes
6answers
2k views

Example of a trigonometric series that is not fourier series?

My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...
0
votes
3answers
98 views

Deducing a $\cos (kx)$ summation from the $e^{ikx}$ summation [duplicate]

I'm trying to solve So far I've done the first part, evaluating the summation ; where a is just n. I'm not sure where to go from here or what it even means deduce the second summation. I ...
1
vote
2answers
44 views

Upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j \; t) \leq f(t,d)$?

I have a sum of a series of trig function as follows: $\sum_{j=1}^{d} cos(2 \pi j \; t)$ where t is just a constant. I am looking for the upper bound $f(t,d)$ such that $\sum_{j=1}^{d} cos(2 \pi j ...
1
vote
1answer
58 views

Why is the Taylor expansion of $\cos$ decreasing?

Why is the Taylor expansion of $\cos$ decreasing ? $\cos(t)=1-\frac{t^2}{2!}+\frac{t^4}{4!}-\frac{t^6}{6!}+...$ such that one can estimate $\cos(t)<1-\frac{t^2}{2!}+\frac{t^4}{4!}$ I ...
34
votes
2answers
506 views

Closed form for $\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$

Please help me to find a closed form for the infinite product $$\prod_{n=1}^\infty\sqrt[2^n]{\tanh(2^n)},$$ where $\tanh(z)=\frac{e^z-e^{-z}}{e^z+e^{-z}}$ is the hyperbolic tangent.
3
votes
2answers
659 views

Does the sum $\sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}$ converge for $\alpha > \frac{1}{2}$ and $x \in [0,2 \pi]$?

Does the series $$ \sum^{\infty}_{k=1} \frac{\sin(kx)}{k^{\alpha}}, $$ converge for all $\alpha > \frac{1}{2}$ and for all $x \in [0,2 \pi]$? It is obvious that it does when $\alpha > 1$, ...
18
votes
1answer
450 views

Proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$

Show that $$\sum_{n=1}^{\infty}\frac{\sinh\big(\pi n\sqrt2\big)-\sin\big(\pi n\sqrt2\big)}{n^3\Big({\cosh\big(\pi n\sqrt2}\big)-\cos\big(\pi n\sqrt2\big)\Big)}=\frac{\pi^3}{18\sqrt2}$$ I have no ...
7
votes
1answer
343 views

Extensions of Ramanujan's Cos/Cosh Identity

The Ramanujan Cos/Cosh Identity is stated here as $$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+ \left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}= ...
8
votes
2answers
177 views

To compute $\tan1-\tan3+\tan5-\cdots+\tan89$, $\tan1+\tan3+\tan5+\cdots+\tan89$

How do we compute : $$i)\ S_1 = \tan1-\tan3+\tan5-\cdots+\tan89$$ and $$ii)\ S_2 = \tan1+\tan3+\tan5+\cdots+\tan89$$ all the angles are in degrees. Thanks
2
votes
2answers
489 views

Evaluation of a trigonometric partial sum

I just wanted to evaluate $$ \sum_{k=0}^n \cos k\theta $$ and I know that it should give $$ \cos\left(\frac{n\theta}{2}\right)\frac{\sin\left(\frac{(n+1)\theta}{2}\right)}{\sin(\theta / 2)} $$ ...
2
votes
2answers
156 views

Positivness of the sum of $\frac{\sin(2k-1)x}{2k-1}$.

For $n\in \mathbb{N}$, $x\in (0,\pi)$. Prove that : $$f_n(x)=\sum_{k=1}^n \frac{\sin [(2k-1)x]}{2k-1} \geq 0.$$ I've tried to do it by differentiation : I Calculate $f_n'(x)$ (sum of ...
4
votes
5answers
111 views

Evaluating $(1+\cos\frac{\pi}{8})(1+\cos\frac{3\pi}{8}) (1+\cos\frac{5\pi}{8})(1+\cos\frac{7\pi}{8})$

How to find the value of $$\left(1+\cos\frac{\pi}{8}\right)\left(1+\cos\frac{3\pi}{8}\right) \left(1+\cos\frac{5\pi}{8}\right)\left(1+\cos\frac{7\pi}{8}\right)$$ I used: $ 1+ \cos ...
6
votes
1answer
88 views

What is $k_{\text{max}}$?

If $[1-\cos x][1 - \cos 2x][1 - \cos 3x] = k\ ; 0º < x < 90º$ Find $k_{\text{max}}$ I have no idea how to solve this I've got $8\left[\sin\left(\frac{x}{2}\right)\times\sin x ...
2
votes
1answer
34 views

Generalization of $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$

I'm looking for a generalization of the following statement $\sup \limits_{\theta} (a \sin \theta + b \cos \theta) = \sqrt{a^2 + b^2}$ In particular, I want to find $\sup \limits_{\theta} (a \sin ...
2
votes
3answers
64 views

Can we subtract a trigonometric term from a polynomial?

Can we find the root of a function like $f(x) = x^2-\cos(x)$ using accurate algebra or do we need to resort to numerical methods approximations? thanks.
1
vote
1answer
102 views

Trigonometry Identities

Consider a collection of five points evenly spaced around a circle to form a regular pentagon. Assume the figure is scaled so that the sides of the pentagon have length 1. Question: Use Ptolemy’s ...