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
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