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

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### Minimize the max value from a sum of sines

I'm considering functions of the form $$\operatorname{f}\left(t\right) = \sin\left(\omega_1 t - \phi_1\right) + \sin\left(\omega_2 t - \phi_2\right) + \cdots + \sin\left(\omega_n t - \phi_n\right)$$ ...
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### Check convergence of a trigonometric series

I have a problem with the following series: $$\sum_{n=1}^{\infty}\sin \frac{1}{\sqrt{n}} \tan \frac{1}{\sqrt{n}}$$ My idea is to check if it's absolutely convergent. As $x\to 0$ we can use following ...
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### Behavior of function $\sum_{j = n}^\infty \frac{\sin^2((2j-1) \pi x)}{(2j-1)^2}$

For a positive integer $n$, define the function $$F_n(x) = n^2 \sum_{j = n}^\infty \frac{\sin^2((2j-1) \pi x)}{(2j-1)^2}.$$ I am trying to understand the behavior of $F_n(x)$ in the following sense. ...
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### $\lim_{p\to\infty}\sum_{n≥1}\arctan\left(\frac{1}{n^p}\right)=\frac{\pi}{4}$

I was solving simple series involving the function $\arctan(x)$ of the form : $$\sum \arctan\left(\frac{d}{ax^2+bx+c}\right)$$ where $a,d,b,c$ are constants. This series can be manipulated too by ...
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### $\lim_{n\rightarrow\infty} \sum_{k=1}^n \text{arccot}(2k^2)$

I think this can be solved using the sandwiching theorem but I have not been able to find appropriate series to sandwich with. The trivial substitution yields that the limit is between $0$ and $\infty$...
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### trigonometric way to solve Sine and Cosine sums? [duplicate]

drawing graphs for most addition or subtractions of Sine and Cosine formula lead to another sine or cosine shaped graph, but I don't know how to actually write them as a formula. For example , drawing ...
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### Solution for $x$, $\sin^{-1} (x-\frac{x^2}{2}+\frac{x^3}{4}-\cdots)+\cos^{-1} (x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots)=\frac{\pi}{2}$

Solution for $x$, $\sin^{-1} (x-\frac{x^2}{2}+\frac{x^3}{4}-\cdots)+\cos^{-1} (x^2-\frac{x^4}{2}+\frac{x^6}{4}-\cdots)=\frac{\pi}{2}$ I have tried, $1-\frac{x^2}{2}+\frac{x^3}{4}-\cdots=:y$ Now, ...
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### Prove $\sum_{n≥1}\frac{\sin n}{n}= \frac{1}{2}(\pi-1)$

I recently learnt that sums of sine and cosine series whose arguments are in arithmetic progression can be evaluated using complex numbers. I modified it a little; like below I divided it by the ...
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### Generalized formula for $\sin((2n-1)x)$?

Does there exist a generalized formula for $\sin((2n-1)x)$? I noticed that if, $\sin(1x)=t^1$ Then $\sin(3x)=3t^1-4t^3$ $\sin(5x)=5t^1-20t^3+16t^5$ $\sin(7x)=7t^1-56t^3+112t^5-64t^7$ $\cdots$ They do ...
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### Prove That an Approximation of $\sin(x)$ via Euler's Formula Approaches $\sin(x)$

I'm trying to approximate the trigonometric functions for a code library, and I want to ask if this is a good way to go about it. I'm aware of the Taylor series approach, but I wanted to go with ...
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