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  "body": "\n## Summary\n\nThe Gauss hypergeometric series is the [[power series|0MZB06ST2B]] $\\sum_{n \\ge 0} \\frac{(a)_n (b)_n}{(c)_n} \\frac{z^n}{n!}$, written ${}_2F_1(a, b; c; z)$, where $(x)_n$ is the rising factorial.\nThe ratio of consecutive coefficients is a rational function of $n$, of degree $2$ over degree $2$.\nMany elementary functions are special cases, and its sum on the unit disc is the starting point of the [[hypergeometric function|463X5V31D9]].\n\n## Explanations\n\n### L1 human\n\nThe geometric series $1 + z + z^2 + \\cdots$ has the same ratio $z$ between all consecutive terms.\nIn the hypergeometric series the ratio of the term with $z^{n+1}$ to the term with $z^n$ is $\\frac{(n+a)(n+b)}{(n+c)(n+1)} z$, which changes with $n$ but tends to $z$.\nSo the series converges for $\\abs{z} \u003c 1$, just like the geometric series.\nWith $a = 1$ and $b = c$ it is exactly the geometric series $1/(1 - z)$.\nWith $a = b = 1$ and $c = 2$ it gives $-\\log(1 - z)/z$.\n",
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