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  "body": "\n## Summary\n\nA trigonometric series is an infinite series $\\sum_{n \\in \\Z} c_n e^{inx}$, or $\\frac{a_0}{2} + \\sum_{n \\ge 1} (a_n \\cos nx + b_n \\sin nx)$, with arbitrary coefficients.\nIts partial sums are [[trigonometric polynomials|J3NJQ5FZJ5]].\nEvery [[Fourier series|G12KXDHPPS]] is a trigonometric series, but not every trigonometric series is the Fourier series of an integrable function.\nThe study of these series led Cantor to set theory, through the question of uniqueness of the coefficients.\n\n## Explanations\n\n### L1 human\n\nA trigonometric series is an infinite sum of waves with whole-number frequencies, each with its own amplitude.\nNothing says in advance that the sum converges or that it comes from a given function.\nFor example, $\\sum_{n \\ge 1} \\frac{\\sin nx}{n^2}$ converges at every $x$, since its terms are bounded by $1/n^2$.\nA Fourier series is the special case where the amplitudes are computed from a function by integrals.\n",
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