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:
“Nature seems at each mans birth to have marked out the bounds of his virtues and vices, and to have determined how good or how wicked that man shall be capable of being.”
—François, Duc De La Rochefoucauld (16131680)
“Thats the great danger of sectarian opinions, they always accept the formulas of past events as useful for the measurement of future events and they never are, if you have high standards of accuracy.”
—John Dos Passos (18961970)