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  "body": "\n## Example\n\nThe series of the positive integers is\n$$\\sum_{n=1}^{\\infty} n = 1 + 2 + 3 + 4 + \\cdots.$$\nIts $N$-th partial sum is $s_N = N(N+1)/2$, which tends to $+\\infty$.\nSo it is a [[divergent series|0JWMJSTBSG]].\n\n## Remarks\n\nIt is not summable by [[Cesàro summation|N8ZGG7EW6J]], since the averages of $s_N$ also tend to $+\\infty$.\nIt is not [[Abel summable|EZXPY4A19C]] either: for $0 \u003c x \u003c 1$, $\\sum_{n \\ge 1} n x^n = x/(1-x)^2$, which tends to $+\\infty$ as $x \\to 1^-$.\n[[Zeta function regularization|VJCJYXGTDY]] assigns it the value $-1/12$.\nFor $\\operatorname{Re} s \u003e 1$, $\\sum_n n^{-s} = \\zeta(s)$, where $\\zeta$ is the [[Riemann zeta function|1XWBY4N0VT]], and the series of the example is formally the case $s = -1$; the analytic continuation of $\\zeta$ has $\\zeta(-1) = -1/12$.\n[[Ramanujan summation|E3SRYCE5VN]] also gives $-1/12$.\nThe value $-1/12$ is not a sum in the usual sense: the partial sums are all positive and grow without bound.\n\n## Explanations\n\n### L1 human\n\nAdding $1 + 2 + 3 + \\cdots$ gives totals that grow without limit, so the series has no ordinary sum.\nThe famous value $-1/12$ comes from a different procedure: one extends the function $\\sum n^{-s}$ beyond the region where the series converges and evaluates it at $s = -1$.\nThis value appears in physics and in number theory, but it should not be read as the result of ordinary addition.\n",
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