Stirling's formula #
This file proves Stirling's formula for the factorial. It states that $n!$ grows asymptotically like $\sqrt{2\pi n}(\frac{n}{e})^n$.
Also some global bounds on the factorial function and the Stirling sequence are proved.
Proof outline #
The proof follows: https://proofwiki.org/wiki/Stirling%27s_Formula.
We proceed in two parts.
Part 1: We consider the sequence $a_n$ of fractions $\frac{n!}{\sqrt{2n}(\frac{n}{e})^n}$ and prove that this sequence converges to a real, positive number $a$. For this the two main ingredients are
- taking the logarithm of the sequence and
- using the series expansion of $\log(1 + x)$.
Part 2: We use the fact that the series defined in part 1 converges against a real number $a$
and prove that $a = \sqrt{\pi}$. Here the main ingredient is the convergence of Wallis' product
formula for π
.
Part 1 #
https://proofwiki.org/wiki/Stirling%27s_Formula#Part_1
Define stirlingSeq n
as $\frac{n!}{\sqrt{2n}(\frac{n}{e})^n}$.
Stirling's formula states that this sequence has limit $\sqrt(π)$.
Instances For
The sequence log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))
has the series expansion
∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))
.
The sequence log ∘ stirlingSeq ∘ succ
is monotone decreasing
We have the bound log (stirlingSeq n) - log (stirlingSeq (n+1))
≤ 1/(4 n^2)
For any n
, we have log_stirlingSeq 1 - log_stirlingSeq n ≤ 1/4 * ∑' 1/k^2
The sequence log_stirlingSeq
is bounded below for n ≥ 1
.
The sequence stirlingSeq
is positive for n > 0
The sequence stirlingSeq
has a positive lower bound.
The sequence stirlingSeq ∘ succ
is monotone decreasing
The limit a
of the sequence stirlingSeq
satisfies 0 < a
Part 2 #
https://proofwiki.org/wiki/Stirling%27s_Formula#Part_2
The sequence n / (2 * n + 1)
tends to 1/2
For any n ≠ 0
, we have the identity
(stirlingSeq n)^4 / (stirlingSeq (2*n))^2 * (n / (2 * n + 1)) = W n
, where W n
is the
n
-th partial product of Wallis' formula for π / 2
.
Suppose the sequence stirlingSeq
(defined above) has the limit a ≠ 0
.
Then the Wallis sequence W n
has limit a^2 / 2
.
Stirling's Formula
Stirling's Formula, formulated in terms of Asymptotics.IsEquivalent
.
Global bounds #
The Stirling sequence is bounded below by √π
, for all positive naturals. Note that this bound
holds for all n > 0
, rather than for sufficiently large n
: it is effective.
Stirling's approximation gives a lower bound for n!
for all n
.
The left hand side is formulated to mimic the usual informal description of the approxmation.
See also factorial_isEquivalent_stirling
which says these are asymptotically equivalent. That
statement gives an upper bound also, but requires sufficiently large n
. In contrast, this one is
only a lower bound, but holds for all n
.
Sharper bounds due to Robbins are available, but are not yet formalised.
Stirling's approximation gives a lower bound for log n!
for all positive n
.
The left hand side is formulated in decreasing order in n
: the higher order terms are first.
This is a consequence of le_factorial_stirling
, but is stated separately since the logarithmic
version is sometimes more practical, and having this version eases algebraic calculations for
applications.
Sharper bounds due to Robbins are available, but are not yet formalised. These would add
lower order terms (beginning with (12 * n)⁻¹
) to the left hand side.