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  "body": "\n## Summary\n\nTwo [[summability methods|8X25Q4XYMN]] are consistent when they never give different values to the same series.\nIf a series is summable by both methods, the two values must agree.\nConsistency is what makes it reasonable to speak of \"the\" generalized sum of a divergent series.\nThe [[Mazur–Orlicz theorem|YE5439NH1T]] gives a condition under which two regular matrix methods are consistent on bounded sequences.\n\n## Explanations\n\n### L1 human\n\nThere are many ways to give a value to a divergent series, such as averaging the partial sums or plugging $x \\to 1$ into a power series.\nOne wants these ways to agree whenever they both work.\nFor the series $1 - 1 + 1 - 1 + \\cdots$, both [[Cesàro summation|N8ZGG7EW6J]] and [[Abel summation|EZXPY4A19C]] give $1/2$.\nIn fact, every series that Cesàro summation sums is summed by Abel summation to the same value, so the two methods are consistent.\n",
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