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  "body": "\n## Definition\n\nFor an integer $r \\ge 0$, the *Favard constant* $K_r$ is the number\n$$K_r = \\frac{4}{\\pi} \\sum_{k=0}^{\\infty} \\left(\\frac{(-1)^k}{2k + 1}\\right)^{r + 1}.$$\n\n## Remarks\n\nThe series converges for every $r \\ge 0$: for $r = 0$ it is an alternating series, and for $r \\ge 1$ it converges absolutely.\n\nExample: $K_0 = \\frac{4}{\\pi} \\left(1 - \\frac{1}{3} + \\frac{1}{5} - \\cdots\\right) = \\frac{4}{\\pi} \\cdot \\frac{\\pi}{4} = 1$.\nLikewise $K_1 = \\frac{4}{\\pi} \\sum_{k \\ge 0} \\frac{1}{(2k+1)^2} = \\frac{4}{\\pi} \\cdot \\frac{\\pi^2}{8} = \\frac{\\pi}{2}$.\nFurther values are $K_2 = \\pi^2/8$ and $K_3 = \\pi^3/24$.\n\nNon-example: $\\frac{4}{\\pi} \\sum_{k \\ge 0} \\frac{1}{2k + 1}$ is not a Favard constant; it lacks the signs of the case $r = 0$ and the series diverges.\n\nThe even-indexed constants increase and the odd-indexed constants decrease, and both tend to $4/\\pi$.\nSo $1 = K_0 \u003c K_2 \u003c K_4 \u003c \\cdots \u003c \\frac{4}{\\pi} \u003c \\cdots \u003c K_3 \u003c K_1 = \\frac{\\pi}{2}$.\n",
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