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26 April 2024
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N 07:56 | Limit to Infinity of Binomial Coefficient over Power/Proof 2 3 changes history +1,600 [CircuitCraft; Prime.mover (2×)] | |||
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07:56 (cur | prev) −39 Prime.mover talk contribs (Removed source citation as it had nothing to do with this result) | ||||
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07:53 (cur | prev) +66 Prime.mover talk contribs | ||||
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03:00 (cur | prev) +1,573 CircuitCraft talk contribs (Created page with "== Theorem == {{:Limit to Infinity of Binomial Coefficient over Power}} == Proof == <onlyinclude> This proof applies to the special case where $k \in \Z$. Then, {{hypothesis}}, we need only consider: :$k \in \set {0, 1, 2, \dotsc}$ By Gamma Function Extends Factorial, it suffices to show: :$\ds \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} = \frac 1 {k !}$ We have: {{begin-eqn}} {{eqn | q = \forall r \in \R | l = \frac {\dbinom r k} {r^k} |...") | |||
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N 07:53 | Limit to Infinity of Binomial Coefficient over Power/Proof 1 2 changes history +1,922 [Prime.mover; CircuitCraft] | |||
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07:53 (cur | prev) +67 Prime.mover talk contribs | ||||
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02:14 (cur | prev) +1,855 CircuitCraft talk contribs (Created page with "== Theorem == {{:Limit to Infinity of Binomial Coefficient over Power}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = \lim_{r \mathop \to \infty} \frac {\dbinom r k} {r^k} | r = \lim_{r \mathop \to \infty} \frac {\map \Gamma {r + 1} } {\map \Gamma {k + 1} \map \Gamma {r - k + 1} r^k} | c = Gamma Function Extends Factorial }} {{eqn | r = \lim_{r \mathop \to \infty} \frac 1 {\map \Gamma {k + 1} } \frac {\sqrt {2 \pi r} \paren {r / e}^r} {\sqrt {2 \p...") | |||
| m 02:16 | Definition:Binomial Coefficient/Real Numbers diffhist 0 CircuitCraft talk contribs | ||||