In books I can find a lot of approximations of factorial, which doesn't require the a lot of multiplications, like for example Stirling's approximation. Very popular is also the Euler's solution which is: $$ n! = \lim_{k \to \infty} \frac{k^n\cdot k!}{(n+1)(n+2)...(n+k)}, \qquad k\in \Bbb N. $$ When can I found explanation how this derive the equation?
2026-03-27 11:39:30.1774611570
Approximation of a factorial
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Following the wikipedia entry on the gamma function, the formula is obtained from $$ n!=n\cdot(n-1)!=n\cdot\Gamma(n). $$
If $k>n$, then $$ \frac{k^n⋅k!}{(n+1)(n+2)...(n+k)}=\frac{k^n⋅k!⋅n!}{(n+k)!}=n!⋅\frac{k^n}{(k+1)(k+2)...(n+k)} $$
and the last fraction, having a fixed number of $n$ factors in numerator and denominator, converges to $1$.