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  "body": "\n## Summary\n\nA lacunary series is a [[trigonometric series|FZK2M8KGXQ]] or a [[power series|0D6KXBHKC0]] whose nonzero terms have frequencies that grow at least geometrically, such as $1, 2, 4, 8, \\ldots$.\nThe large gaps make the terms behave almost like independent random variables.\nAs a result, lacunary series have rigid properties.\nBy the [[Hadamard gap theorem|DA8M8VZZHC]], a power series with such gaps cannot be continued past its circle of convergence.\nA lacunary Fourier series of a bounded function converges absolutely.\n\n## Explanations\n\n### L1 human\n\nMost of the terms of a lacunary series are zero, and the remaining ones are spaced further and further apart.\nFor example, $\\sum_{k \\ge 0} 2^{-k} \\cos(2^k x)$ uses only the frequencies $1, 2, 4, 8, \\ldots$.\nWaves with such widely spaced frequencies hardly interfere with each other.\nThis makes lacunary series a source of both clean theorems and striking examples, such as the nowhere differentiable Weierstrass function.\n",
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  "name": "Lacunary series",
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