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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

$$a_0 + \sum\limits_{n = 1}^{\infty} c_n e^{inx}$$

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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