       Re: and Limit a^n / n!

• To: mathgroup at smc.vnet.net
• Subject: [mg8349] Re: [mg8245] and [mg8130] Limit a^n / n!
• From: Olivier Gerard <jacquesg at pratique.fr>
• Date: Tue, 26 Aug 1997 02:22:43 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```Seanross at worldnett wrote

> agt801 wrote:
> >
> > Hi,
> >
> > Would anyone tell me how to teach mathematica to evaluate the limit :
> >
> > Limit[a ^ n  / n!, n -> Infinity]  where a is a real constant.
> >
> > Leung
>
> Why not graph it as a function of n?  If there is a limit, you will see
> it soon enough.  Don't get hung up on the abstract concept of infinity.
> There are no infinities in nature, only in the human mind when used as a
> useful mathematical abstraction. In the real world, Infinity just means
> larger than you can measure or so large that it becomes the dominant
> signal in your measurement.  I just did this for a few values of a and
> it sure looks like the limit is zero.  At least, 10^100/100! is around
> 10^-175, which is closed enough to zero for me when compared to the
> value of a or n.

Sean, I think the question was not about mathematical knowledge,
in fact this limit is a simple consequence of the Stirling Formula

n!     ->  Sqrt[Pi/2] Sqrt[n] n^n Exp[-n] + ....     (* n real *)
x->oo

Which is one of the oldest (and most useful) asymptotic formula.
The question was, how do I teach Mathematica new Limits I need for
my work ?
I am sure that Mathematica uses in many parts of its code the
Stirling Development (There are terms beyond the principal one I
quoted) but the problem is that when we execute:

Limit[ 4^n / n!, n->Infinity]

we don't reach a result. (On my Mma 3.0.1.0 version, I have got
messages from Series when I do that: Essential Singularity)

In this case Limit relies on the ability of Series to make an
asymptotic expansion of the formula, but Gamma[x] is not your
everyday tame analytic function.

I don't how to teach correctly  Limit and Series to do correctly
their job in this case. These functions are
certainly at the core of many kernel algorithms so when you add
recognition patterns for them you don't know what you loose.

Fortunately, I don't have too. I just load the package:
<<Calculus`Limit

And then the previous expression (as many others involving
PolyGamma for instance) is correctly treated.

That's happy for I routinely use these kinds of expression
in my research. If that is your case you may consider adding
a line in your Init.m to load this package immediately on Kernel boot.

information on this package (and many others).

regards,

Olivier Gerard

```

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