# Math

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Is there a mathematical formula to calculate this easily?

1 x 2 x 3 x 4 x 5...50

I know it's not an arithmetic/geometric series because that means 1 + 2 + 3 + 4 + 5...50

• Math -

There exists a method that allows you to compute this very accurately (and very easily). The formula in this case is:

n! = n^n exp(-n) sqrt(2 pi n)
exp[1/(12 n) - 1/(360 n^3)
+ 1/(1260 n^5) -1/(1680 n^7)+... ]

Here n! = 1 x 2 x 3 x...x n

So, let's test this formula for n = 50. The exact answer you get by multiplying all the factors is:

50! = 3.04140932017..... x 10^(64)

If you evaluate n^n exp(-n) sqrt(2 pi n)
(this is known as Stirling's approximation) for n = 50 you get:

3.03634459394...x 10^64.

The asymptotic series in the exponential is:

1/(12 n) - 1/(360 n^3)
+ 1/(1260 n^5) -1/(1680 n^7)+...

for n = 50 this is
1.66664444699...x10^(-3)

The exponential of this number is
1.00166803407...

3.03634459394...x 10^64 times 1.00166803407... equals 50! to 12 significant figures!

• Math -

1x2=2x3=6x4=24x5=120

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