You first take the log of both sides:
Log(60,000) = Log(n!)
Then you use the asymptotic expansion:
Log(n!) = n Log(n) - n
+ 1/2 Log(2 pi n) + 1/(12 n)
- 1/(360 n^3) + 1/(1260 n^5)- ...
You solve for n using the Newton Raphson method. If we put:
f(n) = Log(n!) then the derivative f'(n) has the expansion:
f'(n) = Log(n)+ 1/(2n)- 1/(12 n^2)
+ 1/(120 n^4) - 1/(252 n^6)+...
If we make some guess for n, then:
n - [f(n) - 60,000]/f'(n)
will be an improved guess which we can improve again using the above formula. A rough starting guess would be
n = Log(60,000) = about 11. The improved value for n will then be 8.34, the next is n = 8.185. Now 8.185! = 60,0031 so this is already pretty accurate!
What is the inverse of a factorial and how would i calculate it.
E.g. 60000 = n!
How would i get n?
1 answer
1 answer