$\qquad\qquad\qquad$ Does $~\displaystyle\lim_{n\to\infty}\frac{\sqrt n}{(-e)^n}\cdot\sum_{k=0}^n\frac{(-n)^k}{k!}~$ possess a closed form expression ?

Inspired by this frequently asked question, I wondered what would happen if the sum were allowed to alternate. Numerically, it seems to converge to a value around $~\dfrac15$ . Unfortunately, I wasn't truly able to grasp any of the various approaches used to evaluate the other related limit $($yes, I actually read carefully through all of them$)$, so I haven't been successful in developing a viable method for expressing this one either. $($Perhaps a new, insightful answer will also help me cast some fresh light on older ones ?$)$

  • $\begingroup$ i'm wondering if one can make such nice probalistic arguments as in the linked question. It seems a little bit tricky because we can't just apply the properties of poisson processes (basicially because $n$ is negative inside the sum). But i'm far from an expert in this field so who knows.. $\endgroup$
    – tired
    Oct 4 '15 at 12:19

It was shown in the answers to this question that

$$ e^{-x}\sum_{k=0}^n\frac{x^k}{k!} = \frac{1}{n!}\int_x^\infty e^{-t}\,t^n\,dt, $$

so setting $x=-n$ we have

$$ \begin{align} e^{n}\sum_{k=0}^n\frac{(-n)^k}{k!} &= \frac{1}{n!}\int_0^\infty e^{-t}\,t^n\,dt + \frac{1}{n!}\int_{-n}^0 e^{-t}\,t^n\,dt \\ &= 1 + \frac{(-1)^n}{n!} \int_0^n e^u u^n\,du \\ &= 1 + \frac{(-1)^n n^{n+1}}{n!} \int_0^1 e^{n [v+\log v]}\,dv. \tag{$*$} \end{align} $$

The quantity $v+\log v$ is increasing and so has a maximum at $v=1$, and near there

$$ v+\log v = 1 + 2(v-1) + O\!\left((v-1)^2\right). $$

By the Laplace method we therefore have

$$ \int_0^1 e^{n [v+\log v]}\,dv \sim \int_{-\infty}^1 e^{n[1 + 2(v-1)]}\,dv = \frac{e^n}{2n}. $$

Using this and Stirling's formula

$$ n! \sim \left(\frac{n}{e}\right)^n \sqrt{2\pi n} $$

we deduce from $(*)$ that

$$ \sum_{k=0}^n\frac{(-n)^k}{k!} \sim \frac{(-e)^n}{2\sqrt{2\pi n}}. $$

The limit in the question is

$$ \lim_{n\to\infty}\frac{\sqrt n}{(-e)^n}\cdot\sum_{k=0}^n\frac{(-n)^k}{k!} = \frac{1}{2\sqrt{2\pi}} = 0.199471\ldots $$

  • $\begingroup$ nice (+1), i was working on the same answer but you beat me for approximatly five minutes! Gratulations! :) $\endgroup$
    – tired
    Oct 4 '15 at 11:44
  • $\begingroup$ Im pretty impressed. I would like to learn more about this Laplace method to solve these problems :) Do you know any good resources? $\endgroup$ Oct 4 '15 at 11:45
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    $\begingroup$ @CalvinKhor I learned it from de Bruijn's Asymptotic Methods in Analysis and Miller's Applied Asymptotic Analysis, both of which are very rigorous and informative. $\endgroup$ Oct 4 '15 at 11:50
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    $\begingroup$ @AntonioVargas theartmad.com/wp-content/uploads/2015/08/… ;) $\endgroup$
    – tired
    Oct 4 '15 at 11:50
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    $\begingroup$ @Lucian I should mention that the poor approximation of $v+\log v$ by $2v-1$ away from $v=1$ doesn't matter when it comes to the limiting behavior. For large $n$ the main contribution to the integral comes from an arbitrarily small neighborhood of the point $v=1$. If $0 < \delta < 1$ then $\int_0^\delta e^{n[v+\log v]}\,dv$ is exponentially smal when compared to $\int_\delta^1 e^{n[v+\log v]}\,dv$. $\endgroup$ Oct 4 '15 at 13:48

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