Re: summing 1/(n!) from 21 to Infinity
- To: mathgroup at smc.vnet.net
- Subject: [mg45022] Re: summing 1/(n!) from 21 to Infinity
- From: "David W. Cantrell" <DWCantrell at sigmaxi.org>
- Date: Sat, 13 Dec 2003 06:06:10 -0500 (EST)
- References: <brci24$2p7$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Sampo Smolander <sampo.smolander+newsnspam at helsinki.fi> wrote:
> I'd be happy if somebody explained what could be behind
> this odd behavior:
>
> When I do:
>
> Sum[ 1 /(n!), {n, 21, Infinity}] // N
>
> I get a -4.44089 * 10^(-16), which doesn't make much
> sense, since it's negative and none of the summands are.
I don't have version 4.0, so I can't confirm that. But version 5 gives 0.,
and so we just need to ask for more accuracy. For example,
N[Sum[1/n!, {n, 21, Infinity}], 10]
gives 2.05...*10^(-20), which is correct.
BTW, Sum[1/n!, {n, 21, Infinity}] gives a correct symbolic answer. But
I'm slightly surprised that I was not able to find a trivial way to get
Mathematica to express that symbolic answer as
E - 6613313319248080001/2432902008176640000
> The same with symbolic starting point,
>
> Sum[ 1 /(n!), {n, m, Infinity}] // N
>
> gives:
>
> E - E Gamma[m,1]/Gamma[m]
I find it hard to believe that ...//N gave you something expressed
literally in terms of E, rather than 2.71828 .
David
> Now where might the mistake be? I don't know enough maths to be able to
> say whether the symbolic sum is wrong -- which however feels more likely
> than a mistake in the implementation of the gamma function.
>
> (I computed the above with Mathematica 4.0, on win98)