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  "body": "\n## Definition\n\nLet $A = (a_{nk})_{n, k \\ge 0}$ be an infinite matrix of real or complex numbers, viewed as a [[matrix summability method|M6311DWJ1E]].\nLet $\\sum_{k \\ge 0} a_k$ be a series with [[partial sums|0NB643K60Q]] $s_k = a_0 + \\dots + a_k$.\nAssume that for every $n$ the series $t_n = \\sum_{k \\ge 0} a_{nk} s_k$ converges.\nThe series $\\sum a_k$ is *absolutely $A$-summable* (or $\\abs{A}$-summable) if\n$$\\sum_{n \\ge 1} \\abs{t_n - t_{n-1}} \u003c \\infty.$$\n\n## Remarks\n\nIf $\\sum_{n \\ge 1} \\abs{t_n - t_{n-1}}$ is finite, then $(t_n)$ is a Cauchy sequence, so it converges.\nThus an absolutely $A$-summable series is $A$-summable.\n\nExample: for the identity matrix, $t_n = s_n$ and $t_n - t_{n-1} = a_n$.\nSo absolute summability by the identity method is [[absolute convergence|0YBDSJARWN]].\n\nNon-example: Grandi's series $1 - 1 + 1 - \\cdots$ is [[Cesàro summable|N8ZGG7EW6J]] to $1/2$, but not absolutely Cesàro summable.\nIts partial sums are $1, 0, 1, 0, \\dots$, so the Cesàro means are $t_{2m} = (m + 1)/(2m + 1)$ and $t_{2m+1} = 1/2$.\nThen $\\abs{t_{2m} - t_{2m-1}} = 1/(2(2m + 1))$, and the sum of these numbers diverges.\n\nFor the Cesàro method the notation is $\\abs{C, 1}$.\nFor Abel summation one says that $\\sum a_n$ is absolutely Abel summable when $x \\mapsto \\sum a_n x^n$ has bounded variation on $[0, 1)$.\n",
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