How to evaluate factorials greater than $69!$?
On my calculator, $69!$ is the largest number I can enter before it gives me a syntax error, most likely due to an overflow.
Is there a way to evaluate huge numbers such as $493!$ by hand? I understand if this method exists (which it probably does, I hope.) that it might be very complex, but any answer is appreciated. Thanks in advance. (To 3 decimals places.)
Best Answer
Well, you can quickly get a very accurate calculation $n!$ by Stirling's approximation:
$$ n! \approx \sqrt{2\pi n} \left( \frac{n}{e} \right) ^n \left( 1 + \frac{1}{12n} + \frac{1}{288 n^2} \right) $$ For $n = 439$ this is accurate to one part in 50 billion.
Your big headache is representing that number on your calculator or in a float number of a computer. That is why people working with factorials of large numbers frequently work with $\log (n!)$.
The answer, by the way, works out to $$1.279533 \cdot 10^{971} $$