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:
“Prohibition will work great injury to the cause of temperance. It is a species of intemperance within itself, for it goes beyond the bounds of reason in that it attempts to control a mans appetite by legislation, and makes a crime out of things that are not crimes. A Prohibition law strikes a blow at the very principles upon which our government was founded.”
—Abraham Lincoln (18091865)
“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)