2
Checking some sums I found empirically a sum which is like the Stirling formula for to $n!$
Conjecture :
Let $k$ be a positive integer greater than $10$. Then it seems we have
$$|-k!+\sum_{n=1}^{\infty}n^{k}e^{-n}|<|\sqrt{2k\pi}\left(\frac{k}{e}\right)^{k}-k!|$$
For small values the differenc...
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of
n
{\displaystyle n}
. It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.One way of stating the approximation involves the logarithm of the factorial:
where the big O notation means that, for all sufficiently large values of
n
{\displaystyle n}
, the difference between...
