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  "body": "\n## Definition\n\nLet $\\sum_{n \\ge 0} a_n$ be a series of complex numbers with [[partial sums|0NB643K60Q]] $s_n = a_0 + \\cdots + a_n$.\n\n1. The series is *Cesàro summable* to $s$, or $(C, 1)$ summable to $s$, if the [[Cesàro means|0P0DJHBGBB]] of the partial sums converge to $s$:\n$$\\sigma_n = \\frac{s_0 + s_1 + \\cdots + s_n}{n + 1} \\to s \\quad (n \\to \\infty).$$\n2. More generally, let $\\alpha \u003e -1$ be real, and put $A^{\\alpha}_0 = 1$ and $A^{\\alpha}_m = \\frac{(\\alpha + 1)(\\alpha + 2) \\cdots (\\alpha + m)}{m!}$ for $m \\ge 1$.\nThe series is $(C, \\alpha)$ summable to $s$ if\n$$\\sigma^{\\alpha}_n = \\frac{1}{A^{\\alpha}_n} \\sum_{k=0}^{n} A^{\\alpha}_{n-k} a_k \\to s \\quad (n \\to \\infty).$$\n\n## Remarks\n\nFor $\\alpha = 0$ one has $\\sigma^0_n = s_n$, which is ordinary convergence.\nFor $\\alpha = 1$ one has $A^1_m = m + 1$, and $\\sigma^1_n = \\frac{1}{n+1} \\sum_{k=0}^{n} (n - k + 1) a_k = \\sigma_n$, so clause 2 extends clause 1.\n\nExample: for $1 - 1 + 1 - \\cdots$ the partial sums are $1, 0, 1, 0, \\dots$.\nSo $\\sigma_{2m} = \\frac{m+1}{2m+1}$ and $\\sigma_{2m+1} = \\frac{1}{2}$, and both tend to $1/2$.\nThe series is $(C, 1)$ summable to $1/2$.\n\nNon-example: for $1 - 2 + 3 - 4 + \\cdots$ the partial sums are $1, -1, 2, -2, 3, \\dots$.\nTheir sums are $1, 0, 2, 0, 3, 0, \\dots$, so $\\sigma_{2m} = \\frac{m+1}{2m+1} \\to \\frac12$ and $\\sigma_{2m+1} = 0$.\nThe means oscillate, so the series is not $(C, 1)$ summable.\nIt is $(C, 2)$ summable to $1/4$.\n\nIf a series is $(C, \\alpha)$ summable to $s$, it is $(C, \\beta)$ summable to $s$ for every $\\beta \u003e \\alpha$.\n",
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