Factorial Fac=
ts
Definition: n! =3D
n(n-1)(n-2)…(3)(2)(1)
Examples:
6! =3D (6)(5)(4)(3)(2)(1)=3D720
3! =3D (3)(2)(1)=3D6
We define 0!=3D1, which doesn’t seem to make sen=
se, but
it works well in formulas such as Taylor
polynomial.
(n+1)! =3D (n+1) n!&n=
bsp;
from the definition – work it out!
So ![](3D"FactorialFacts_files/image002.gif")
Another example ![](3D"FactorialFacts_files/image004.gif")
Limits with F=
actorials
Factorial n! grows much faster than any exponential su=
ch as ![](3D"FactorialFacts_files/image006.gif")
or
![](3D"FactorialFacts_files/image008.gif")
. The expon=
ential
may be larger at first, but the factorial will overtake it and grow much fa=
ster
as n approaches infinity.
Therefore, ![](3D"FactorialFacts_files/image010.gif")
and
![](3D"FactorialFacts_files/image012.gif")
for any fix=
ed
number a.
However ![](3D"FactorialFacts_files/image014.gif")
grows much =
faster
than n! as n approaches infinity.
So ![](3D"FactorialFacts_files/image016.gif")
and
![](3D"FactorialFacts_files/image018.gif")
.
Order of Grow=
th: , n!, exponen=
tial,
polynomial (with higher order polynomials growing faster).
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