Factorial as Sum of Series of Subfactorial by Falling Factorial over Factorial

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Theorem

\(\displaystyle n!\) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \dfrac { {!k} \, n^{\underline k} } {k!}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac { !0 \times n^{\underline 0} } {0!} + \dfrac { {!1} \times n^{\underline 1} } {1!} + \dfrac { {!2} \times n^{\underline 2} } {2!} + \dfrac { {!3} \times n^{\underline 3} } {3!} + \cdots\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + \left({1 - \dfrac 1 {1 !} }\right) n + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} }\right) n \left({n - 1}\right) + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} - \dfrac 1 {3 !} }\right) n \left({n - 1}\right) \left({n - 2}\right) + \cdots\)


Here, we have:

$n^{\underline k}$ denotes the $k$th falling factorial of $n$
$!k$ denotes the $k$th subfactorial
$k!$ denotes the $k$th factorial.


Proof

Let $n$ be a non-negative integer.

We assume a solution of the form:

$(1): \quad n! = a_0 + a_1 n + a_2 n \left({n - 1}\right) + a_3 n \left({n - 1}\right) \left({n - 2}\right) + \cdots$


We can express $(1)$ using binomial coefficients:

$(2): \quad n! = \displaystyle \sum_k \dbinom n k k! a_k$


Then:

\(\displaystyle \sum_n n! \binom m n \left({-1}\right)^{n - k}\) \(=\) \(\displaystyle \sum_n \left({\sum_k \binom n k k! \, a_k}\right) \binom m n \left({-1}\right)^{n - k}\) substituting for $n!$ from $(2)$
\(\displaystyle \) \(=\) \(\displaystyle \sum_k k! \, a_k \sum_n \binom n k \binom m n \left({-1}\right)^{m - n}\) changing order of summation
\(\displaystyle \) \(=\) \(\displaystyle \sum_k k! \, a_k \delta_{k m}\) Corollary to Sum over $k$ of $\dbinom r k \dbinom {s + k} n \left({-1}\right)^{r - k}$
\(\displaystyle \) \(=\) \(\displaystyle m! \, a_m\) as the $m$th term is the only one left standing
\(\displaystyle \leadsto \ \ \) \(\displaystyle a_m\) \(=\) \(\displaystyle \sum_{n \mathop \ge 0} \left({-1}\right)^{m - n} \dfrac {n!} {m!} \binom m n\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^m \left({-1}\right)^{m - n} \dfrac {n!} {m!} \dfrac {m!} {n! \, \left({m - n}\right)!}\) Definition of Binomial Coefficient
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^m \frac {\left({-1}\right)^{m - n} } {\left({m - n}\right)!}\) simplifying
\(\displaystyle \) \(=\) \(\displaystyle \sum_{n \mathop = 0}^m \frac {\left({-1}\right)^n} {n!}\) Permutation of Indices of Summation

$\blacksquare$


Condition for Convergence

Consider the series:

\(\displaystyle n!\) \(=\) \(\displaystyle \sum_{k \mathop \ge 0} \dfrac { {!k} \, n^{\underline k} } {k!}\)
\(\displaystyle \) \(=\) \(\displaystyle \dfrac { !0 \times n^{\underline 0} } {0!} + \dfrac { {!1} \times n^{\underline 1} } {1!} + \dfrac { {!2} \times n^{\underline 2} } {2!} + \dfrac { {!3} \times n^{\underline 3} } {3!} + \cdots\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + \left({1 - \dfrac 1 {1 !} }\right) n + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} }\right) n \left({n - 1}\right) + \left({1 - \dfrac 1 {1 !} + \dfrac 1 {2 !} - \dfrac 1 {3 !} }\right) n \left({n - 1}\right) \left({n - 2}\right) + \cdots\)


This converges only when $n \in \Z_{\ge 0}$, that is, when $n$ is a non-negative integer.


Historical Note

This result was established by James Stirling during the course of his attempt extend the factorial to the real numbers.

While it provides an exact value for $n!$ when $n$ is a positive integer, the infinite sum exists only under this condition.


Sources