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  "body": "\n## Example\n\n*Grandi's series* is the series\n$$\\sum_{n=0}^{\\infty} (-1)^n = 1 - 1 + 1 - 1 + \\cdots.$$\nIts partial sums are $s_N = \\sum_{n=0}^{N} (-1)^n$, which equal $1$ for even $N$ and $0$ for odd $N$.\nSo the partial sums have no limit, and the series is a [[divergent series|0JWMJSTBSG]].\nIts terms do not tend to $0$, which already rules out convergence.\n\n## Remarks\n\nSeveral summation methods assign it the value $1/2$.\n[[Cesàro summation|N8ZGG7EW6J]]: the averages $(s_0 + \\cdots + s_N)/(N+1)$ equal $1/2$ for odd $N$ and $(N/2 + 1)/(N+1)$ for even $N$, so they tend to $1/2$.\n[[Abel summation|EZXPY4A19C]]: for $0 \u003c x \u003c 1$, $\\sum_{n \\ge 0} (-1)^n x^n = 1/(1 + x)$, which tends to $1/2$ as $x \\to 1^-$.\n[[Borel summation|8DJPSYSHGJ]] also gives $1/2$, since it sums the geometric series $\\sum z^n$ to $1/(1 - z)$ at $z = -1$.\nThe value $1/2$ is forced for any linear and stable method that sums the series: if $S = 1 - 1 + 1 - \\cdots$, stability and linearity give $S = 1 - S$.\n\n## Explanations\n\n### L1 human\n\nAdd $1$, subtract $1$, add $1$, subtract $1$, forever.\nThe running total jumps between $1$ and $0$ and never settles, so the series has no sum in the ordinary sense.\nOn average the running total is $1/2$, and this is the value that the usual summation methods give it.\n",
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