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  "body": "\n## Definition\n\nLet $A$ and $B$ be [[summability methods|8X25Q4XYMN]] for series of real or complex numbers.\nFor a series $\\sum a_n$ that is $A$-summable, write $A\\text{-}\\sum a_n$ for the value that $A$ assigns to it, and similarly for $B$.\nThe methods $A$ and $B$ are *consistent* if for every series $\\sum a_n$ that is both $A$-summable and $B$-summable,\n$$A\\text{-}\\sum a_n = B\\text{-}\\sum a_n.$$\n\n## Remarks\n\nExample: Cesàro summation and Abel summation are consistent.\nIndeed, if $\\sum a_n$ is Cesàro summable to $s$, then it is Abel summable to the same $s$, so whenever both methods apply their values agree.\n\nNon-example: let $A$ assign to a series the limit of its even [[partial sums|0NB643K60Q]] $s_0, s_2, s_4, \\dots$, and let $B$ assign the limit of its odd partial sums $s_1, s_3, s_5, \\dots$.\nBoth methods give the usual sum to every convergent series.\nFor $1 - 1 + 1 - 1 + \\cdots$, the even partial sums are all $1$ and the odd ones are all $0$.\nSo both methods sum this series, with values $1$ and $0$, and $A$ and $B$ are not consistent.\n\nThe same notion applies to methods that act on sequences rather than series, by comparing the values they assign to a common summable sequence.\n",
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