Questions tagged [trigonometric-series]

For questions about or related to trigonometric series, i.e. series of the form $a_0 + \sum_{n = 1}^{\infty} (a_n \cos{nx} + b_n \sin{nx})$ or $\sum_n c_n e^{inx}$.

A trigonometric series is any series of the form

$$a_0 + \sum\limits_{n = 1}^{\infty} \Big(a_n \cos{nx} + b_n \sin{nx}\Big)$$

with $a_n$ and $b_n$ complex numbers. Using Euler's identity, any trigonometric series can also be written in the form

$$\sum\limits_{n = -\infty}^{\infty} c_n e^{inx}$$

If the coefficients $a_n$ and $b_n$ can be evaluated by

\begin{align*} \pi a_n &= \int_0^{2\pi} f(x) \cos{nx} dx \\ \pi b_n &= \int_0^{2\pi} f(x) \sin{nx} dx \end{align*}

with $f$ an integrable function, then the series is called a Fourier series.

It is known that if a trigonometric series converges to a function on $[0, 2\pi]$ which is zero (except at at-most finitely many points), then every coefficient $a_n$ and $b_n$ must be zero.

Note that many authors define trigonometric series to be $1$-periodic by considering the interval $[0, 1]$ and replacing $n$ with $2\pi n$.

Reference: Trigonometric series.

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A Nice Problem In Additive Number Theory

$\color{red}{\mathrm{Problem:}}$ $n\geq3$ is a given positive integer, and $a_1 ,a_2, a_3, \ldots ,a_n$ are all given integers that aren't multiples of $n$ and $a_1 + \cdots + a_n$ is also not a multiple of $n$. Prove there are at least $n$…
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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.
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Is $\sum_{a=0}^m\sum_{b=0}^n\cos(abx)$ always positive?

Fix integers $m,n\geq0$. Do we have the inequality $\displaystyle\sum_{a=0}^m\sum_{b=0}^n\cos(abx)>0$ for all $x\in\mathbb{R}$? We can also write this function…
Thomas Browning
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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 hint as to how to even start.
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Finite sum $\sum_{n=2}^N\frac{1}{n^2}\sin^2(\pi x)\csc^2(\frac{\pi x}{n})$

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 found this function that has very interesting…
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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\left(\theta + \frac{2\pi}{n}\right) + \dots + \tan \left (\theta + \frac{(n-1)\pi}{n} \right) = -n\cot…
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Asymptotic behaviour of sum

I would like to evaluate the number $c$ given by $$ c = \lim_{m\to\infty} \frac{1}{\log m}\sum_{n=1}^m \frac{1}{n^2 \sin^2(\pi n \tau)} $$ where $\tau = (1+\sqrt{5})/2$. My attempt: my guess was this sum would be dominated by the terms for which…
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Consecutively Adding Sines

One thing, I'm not a mathematician so please be patient. I am still in Algebra II Trig. Leading with that, why does $$ x_0 = \sin 1, \space x_1 = x_0 + \sin x_0, \space x_2 = x_1 + \sin x_1 ... $$ and after a while, $$ x = \pi $$ I know this to be…
Corpus Shmorpus
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A Sine and Inverse Sine integral

A demonstration of methods While reviewing an old text book an integral containing sines and sine inverse was encountered, namely, $$\int_{0}^{\pi/2} \int_{0}^{\pi/2} \sin(x) \, \sin^{-1}(\sin(x) \, \sin(y)) \, dx \, dy = \frac{\pi^{2}}{4} -…
Leucippus
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Closed form for $\sum_{n=1}^\infty\frac{\cos(\pi \log n)}{n^2}$

Is there a closed form for the following sum? $$\sum_{n=1}^\infty\frac{\cos(\pi\log n)}{n^2}$$
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What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
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How to evaluate $\sum\limits_{k=0}^{n-1} \sin^t(\pi k/2n)$?

How to evaluate $\displaystyle\sum_{k=0}^{n-1} \sin^t\left(\frac{\pi k}{2n}\right)$? $t,n$ are constants $\in \Bbb{Z}$. My try: $$\begin{align} \zeta:=e^{i\pi/2n} \implies & \sum_{k=0}^{n-1}\sin^t\left(\frac{\pi k}{2n}\right) \\ =…
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Why does a fourier series have a 1/2 in front of the a_0 coefficient

I am reading up on the fourier series, and I keep seeing it as being defined as: $$ f(\theta)= \frac{1}{2}a_0 + \sum_{n=1}^{\infty}(a_n \cos(n\theta) + b_n \sin(n\theta)) $$ where $$ a_n =…
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Extensions of Ramanujan's Cos/Cosh Identity

The Ramanujan Cos/Cosh Identity is stated here as $$\left[1+2\sum_{n=1}^{\infty}\frac{\cos n\theta}{\cosh n\pi}\right]^{-2}+ \left[1+2\sum_{n=1}^{\infty}\frac{\cosh n\theta}{\cosh n\pi}\right]^{-2}= \frac{2\Gamma^4\left(\frac34\right)}{\pi}$$ Then…
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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: $$\sum_{n=0}^\infty\frac{1}{2^n}\tan\frac{1}{2^n}$$
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