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  "body": "\n## Definition\n\nLet $(c_n)_{n \\in \\Z}$ be a sequence of complex numbers.\nThe *trigonometric series* with coefficients $c_n$ is the formal [[series|0NM626FH6Q]] of functions of $x \\in \\R$\n$$\\sum_{n=-\\infty}^{\\infty} c_n e^{inx},$$\nwhere $e^{inx}$ is the [[complex exponential|07GJV36896]].\nIts $N$-th partial sum is the [[trigonometric polynomial|J3NJQ5FZJ5]] $S_N(x) = \\sum_{n=-N}^{N} c_n e^{inx}$.\nThe series *converges* at $x$ when $\\lim_{N \\to \\infty} S_N(x)$ exists.\n\n## Remarks\n\nIn real form the series is $\\frac{a_0}{2} + \\sum_{n=1}^{\\infty} (a_n \\cos nx + b_n \\sin nx)$, with $a_n = c_n + c_{-n}$ and $b_n = i(c_n - c_{-n})$, and its partial sums are the same.\nExample: $\\sum_{n=1}^{\\infty} \\frac{\\cos nx}{n^2}$ is a trigonometric series with $c_0 = 0$ and $c_n = c_{-n} = \\frac{1}{2n^2}$ for $n \\ge 1$; it converges at every $x$ by comparison with $\\sum 1/n^2$.\nNon-example: the power series $\\sum_{n \\ge 0} x^n$ is a series of functions, but its terms are powers of $x$ and not multiples of $e^{inx}$, so it is not a trigonometric series.\nA trigonometric series is the [[Fourier series|G12KXDHPPS]] of an integrable function only for special coefficients.\nFor instance, $\\sum_{n \\ge 2} \\frac{\\sin nx}{\\log n}$ converges at every $x$ but is not the Fourier series of a Lebesgue integrable function.\n",
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