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  "body": "\n## Statement\n\nFor every real $x$, the series $\\sum_{n=0}^{\\infty} \\frac{(-1)^n x^{2n}}{(2n)!}$ converges absolutely, and\n$$\\cos x = 1 - \\frac{x^2}{2!} + \\frac{x^4}{4!} - \\cdots.$$\nThis series is the [[Maclaurin series|043SJ5RF59]] of the cosine function.\n\n## Remarks\n\nThe [[ratio of the terms|0JBMYAA9QH]] is $\\frac{x^2}{(2n+1)(2n+2)} \\to 0$. So the radius of convergence is $\\infty$.\nFor $|x| \\le 1$, the terms decrease in size. So the partial sums lie alternately above and below $\\cos x$.\nMathlib states the series as `Real.hasSum_cos`.\n\n## Explanations\n\n### L1 human\n\nCosine is an infinite even polynomial with alternating signs.\nNear $0$, $\\cos x$ is close to $1 - x^2/2$.\n",
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