Bounds and Asymptotic Formulas
The following bounds for hold:
Stirling's approximation yields the bounds:
- and, in general, for m ≥ 2 and n ≥ 1,
and the approximation
- as
The infinite product formula (cf. Gamma function, alternative definition)
yields the asymptotic formulas
as .
This asymptotic behaviour is contained in the approximation
as well. (Here is the k-th harmonic number and is the Euler–Mascheroni constant).
The sum of binomial coefficients can be bounded by a term exponential in and the binary entropy of the largest that occurs. More precisely, for and, it holds
where is the binary entropy of .
A simple and rough upper bound for the sum of binomial coefficients is given by the formula below (not difficult to prove)
Read more about this topic: Binomial Coefficient
Famous quotes containing the words bounds and/or formulas:
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—François, Duc De La Rochefoucauld (1613–1680)
“You treat world history as a mathematician does mathematics, in which nothing but laws and formulas exist, no reality, no good and evil, no time, no yesterday, no tomorrow, nothing but an eternal, shallow, mathematical present.”
—Hermann Hesse (1877–1962)