# 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 $\sum_n c_n e^{inx}$.

780 questions
Filter by
Sorted by
Tagged with
47 views

20 views

### Explicit expression for a recursive sequence containing a trigonometric function

I would like to determine the explicit function of a recursive sequence. The recursive sequence is: $$x_{n+1} = \tanh(x_n)$$ I am not sure how to approach this problem. In another post I saw that ...
29 views

46 views

### If $x\cot x=a_0+a_2x^2+a_4x^4+\cdots$, then $\frac{a_{2n}}{1!}-\frac{a_{2n-1}}{3!}+\cdots+\frac{(-1)^na_0}{(2n+1)!}=\frac{(-1)^n}{(2n)!}$

If $x\cot x=a_0+a_2x^2+a_4x^4+\cdots$, then show that $$\dfrac{a_{2n}}{1!}-\dfrac{a_{2n-2}}{3!}+\dfrac{a_{2n-4}}{5!}-\cdots+\dfrac{(-1)^na_{0}}{(2n+1)!}=\dfrac{(-1)^n}{(2n)!}$$ This problem is from ...
108 views

### Inequality regarding L^6 norm of a trigonometric polynomial

I am reading Bourgain's paper "Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations", and am stuck on the following ...
73 views

46 views

### Theorem 8.14 in Rudin's “Principles of Mathematical Analysis”

I am reading Rudin's "Principles of Mathematical Analysis" and I am stuck at the following theorem Theorem 8.14: If, for some $x$, there are constants $\delta > 0$ and $M < \infty$ ...
34 views

### Divisibility of Chebyshev Polynomials

I was trying to solve a problem involving an Insect crawling on the Cartesian/Coordinate Plane. We have an insect on the origin of the coordinate plane, who remembers a particular angle $\theta.$ We ...
55 views

### Solving a nonlinear first order ODE

In this text https://www.springer.com/gp/book/9781461454762 on p. 95, it has the following The text does not explain how to get to this solution. $B, C, D$ are arbitrary constants. With $A>0$, I ...
59 views

22 views

### Determine the convergence or divergence of the series

I'm having some trouble trying to get started on this problem. $\sum_{n=1}^{\infty} [\sin(1/(2n)) − \sin(1/(2n+1))]$. My initial thoughts were to try to work the series as two individual series, see ...
39 views

How does the author of this book make these simplifications from left to right? It doesn’t seem obvious. Here $h,\alpha$ are real numbers and $N$ is a natural number $$\left|\sum_{n=1}^{N}e^{2\pi i h ... 0answers 156 views ### Does this erratically-behaved infinite sum converge? I have been trying to find the convergence (and value) of this infinite sum:$$\sum_{n=1}^\infty \frac{\sin(n)^n}{\cos(en)} The partial sums behave relatively unpredictably and at some point become ...
I'm working on a problem in acoustic scattering, where I have to calculate the torque exerted on a cylinder. The torque, $\tau(\phi,m)$, is a function of the "maximum phase" denoted by $\phi$...