How fast does $\frac{1}{n!} \sum_{k=0}^n k(n-k)!$ grow, as a function of $n$?

Question

Consider the sum $$\frac{1}{n!} \sum_{k=0}^n k(n-k)!$$ How fast does it grow (as a function of $n$)? I can prove that it grows slower than or equal to linearly (but I expect this is very crude), so I am looking for a better bound of growth.

The closed form I found in Wolfram Alpha includes subtracting some factorials, so is not a keen indicator of growth. I tried bounding this using the Gamma function: It is $\le \int_0^n x\Gamma(n-x+1)$, but I am not sure what I can do from here.

Where does this crop up? Well, it is the expected stopping time to a common "puzzle" question.

Answer 1

This sum is bounded above, and it's easy to show a crude estimate. Let

$$f_n(x) = \sum_{k=0}^n\,\frac{(n-k)!}{n!}\,x^k=\sum_{k=0}^n\,\frac{1}{\binom{n}k}\,\frac{x^k}{k!}.$$

Then

\begin{align} f_n'(x) = \sum_{k=0}^n\,\frac{k\,(n-k)!}{n!}\,x^k =\sum_{k=1}^n\,\frac{1}{\binom{n}k}\,\frac{x^{k-1}}{(k-1)!} &\leq \sum_{k=1}^n\,\frac{x^{k-1}}{(k-1)!}\\ &= \sum_{k=0}^{n-1}\,\frac{x^{k}}{k!}\\ &\leq \sum_{k=0}^{\infty}\,\frac{x^{k}}{k!}=\exp(x), \end{align}

and observe that this bound does not depend on $n$. Moreover, notice that $\frac{1}{n!} \sum_{k=0}^n k(n-k)! =$ $f'(1)$. It follows that

$$\frac{1}{n!} \sum_{k=0}^n k(n-k)! \leq e$$

Answer 2

As shown in the table below, the function $$f_n=\frac{1}{n!} \sum_{k=0}^n k\,(n-k)!$$ is decreasing as soon as $n>2$ $$\left( \begin{array}{ccc} n & f_n & f_n\approx \\ 0 & 0 & 0.00000 \\ 1 & 1 & 1.00000 \\ 2 & \frac{3}{2} & 1.50000 \\ 3 & \frac{7}{6} & 1.16667 \\ 4 & \frac{17}{24} & 0.70833 \\ 5 & \frac{17}{40} & 0.42500 \\ 6 & \frac{41}{144} & 0.28472 \\ 7 & \frac{1079}{5040} & 0.21409 \\ 8 & \frac{111}{640} & 0.17344 \\ 9 & \frac{53227}{362880} & 0.14668 \\ 10 & \frac{462341}{3628800} & 0.12741 \end{array} \right)$$ For the interval $2 \leq n \leq 100$, it seems that is varies "almost" as $$f_n=\frac{a+b n}{1+c n + d n^2}$$ A quick and dirty nonlinear regression gives $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} \\ a & +0.106334 & 0.029581 \\ b & +0.155564 & 0.005736 \\ c & -0.740640 & 0.010730 \\ d & +0.189945 & 0.002423 \\ \end{array}$$

Edit

After Jack D'Aurizio's answer, which clearly shows the limit, we can get the expression for $f_n$ since $$\int_{0}^{+\infty}(n-k)x^k e^{-x}\,dx=(n-k)\, \Gamma (k+1)$$ making $$\sum_{k=0}^{n}(n-k)k! =-n\, \text{Subfactorial}[-1]-(-1)^n n\, \Gamma (n+2)\, \text{Subfactorial}[-n-2]-\Gamma (n+2)+1$$ where appears the subfactorial function and, as Jack D'Aurizio showed, this is $\ll (n-1)!$.

Answer 3

We can rewrite the series as $$\sum_{k = 0}^n k(n-k)! = \sum_{k = 0}^n (n-k)k! = n\sum_{k = 0}^n k! - \sum_{k=0}^n k \cdot k! = n\sum_{k = 0}^n k! - (n+1)!+1$$ where the last equality follows from the identity $\sum_{i = 0}^n k \cdot k! = (n+1)! - 1$. If this post is to be believed, we have an asymptotic expansion

$$ \sum_{k = 0}^n k! = n!\left(1 + \frac{1}{n} + \frac{1}{n^2} + O\left(\frac{1}{n^3}\right)\right)$$

Using these two observations to your series gives $$\frac{1}{n!}\sum_{k=0}^nk(n-k)! = n\left(1 + \frac{1}{n} + \frac{1}{n^2} + O\left(\frac{1}{n^3}\right)\right) - (n+1)+\frac{1}{n!} = \frac{1}{n} + O\left(\frac{1}{n^2}\right)$$

And hence as $n \to \infty$, the series goes to $0$.

(Disclaimer: I don't really know much about asymptotic series, so some of my reasoning there may be off).

Answer 4

$$\sum_{k=0}^{n}(n-k)k! = \int_{0}^{+\infty}\sum_{k=0}^{n}(n-k)x^k e^{-x}\,dx =\int_{0}^{+\infty}\underbrace{\frac{n-(n+1)x+x^{n+1}}{(1-x)^2}}_{\ll x^{n-1}}e^{-x}\,dx$$ hence the LHS is $\ll (n-1)!$ and the wanted limit is clearly zero.