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  "body": "\n## Example\n\nThe series\n$$\\sum_{n=1}^{\\infty} 1 = 1 + 1 + 1 + 1 + \\cdots$$\nhas $N$-th partial sum $s_N = N$, which tends to $+\\infty$.\nSo it is a [[divergent series|0JWMJSTBSG]].\n\n## Remarks\n\nThe [[Cesàro means|N8ZGG7EW6J]] $(s_1 + \\cdots + s_N)/N = (N+1)/2$ also tend to $+\\infty$.\nFor $0 \u003c x \u003c 1$, $\\sum_{n \\ge 1} x^n = x/(1-x)$, which tends to $+\\infty$ as $x \\to 1^-$, so the series is not [[Abel summable|EZXPY4A19C]].\n[[Zeta function regularization|VJCJYXGTDY]] assigns it the value $-1/2$.\nIndeed $\\sum_n n^{-s} = \\zeta(s)$ for $\\operatorname{Re} s \u003e 1$, the series of the example is formally the case $s = 0$, and the analytic continuation of the [[Riemann zeta function|1XWBY4N0VT]] has $\\zeta(0) = -1/2$.\nNo linear and stable method can give it a finite value $S$: stability gives $S = 1 + S$, which is impossible.\n\n## Explanations\n\n### L1 human\n\nAdding $1$ forever gives totals $1, 2, 3, \\ldots$ that grow without bound.\nThis is the simplest divergent series.\nThe value $-1/2$ that sometimes appears for it comes from the Riemann zeta function at $0$, not from ordinary addition.\n",
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