Problem 11: What are these functions?
Calculus
Level
2
!n, also known as the derangement function, shows you the number of permutations that allow no one to get a correct item.
If !n is found by:
!n=n!(1/2-1/6+1/24-...+(-1)^n/n!)
find the limit of n to infinity of n!/!n
sqrt2
pi
sqrt3
e
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1 solution
One way to solve this problem is no notice that the coefficient of n! in the original equation is one of the Maclaurin Series for e^x. Because the terms oscillate back and forth from positive to negative, and there is only a coefficient of 1 (other than the 1/p! on the bottom), we can conclude that x is -1 (when n goes to infinity).
So: !n=n!/e
n!/!n=e