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  "body": "\n## Summary\n\nRamanujan summation assigns a value to a series $\\sum_{n \\ge 1} f(n)$ by taking the constant term in the [[Euler–Maclaurin formula|9P839VB8S7]] for its partial sums.\nThe value is expressed through $f(0)$, the odd derivatives of $f$ at $0$ and the [[Bernoulli numbers|0KC00K7KY4]].\nIt gives $1 + 2 + 3 + \\cdots = -\\frac{1}{12}$ and $1 + 1 + 1 + \\cdots = -\\frac12$, the values $\\zeta(-1)$ and $\\zeta(0)$ of the [[Riemann zeta function|1XWBY4N0VT]].\nUnlike most [[summability methods|8X25Q4XYMN]], it is neither regular nor stable under shifting the index.\n\n## Explanations\n\n### L1 human\n\nThe partial sums $f(1) + \\cdots + f(N)$ of a series often behave like an integral of $f$ plus correction terms plus a constant.\nRamanujan proposed to call that constant the \"sum\" of the series, even when the series diverges.\nFor $f(n) = n$, the partial sums are $\\frac{N^2}{2} + \\frac{N}{2}$ exactly, and the constant obtained by his rule is $-\\frac{1}{12}$.\nThis number does not mean that the positive integers add up to a negative number; it is a different kind of value, attached to the function $f$.\n",
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