Re: summing 1/(n!) from 21 to Infinity
- To: mathgroup at smc.vnet.net
- Subject: [mg45038] Re: [mg44998] summing 1/(n!) from 21 to Infinity
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Sat, 13 Dec 2003 06:06:53 -0500 (EST)
- References: <200312120941.EAA24153@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 12 Dec 2003, at 18:41, Sampo Smolander 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.
>
> The same with symbolic starting point,
>
> Sum[ 1 /(n!), {n, m, Infinity}] // N
>
> gives:
>
> E - E Gamma[m,1]/Gamma[m]
>
> 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)
>
>
It does make sense, since the number is so small that with the
requested precision you can't expect anythng better. In fact
Mathematica 5.0 gives you a somewhat differnet answer:
N[Sum[1/n!, {n, 21, Infinity}]]
0.
If you want a more accuarate numerical answer you must use more
precision:
N[Sum[1/n!, {n, 21, Infinity}], 20]
Out[5]=
2.050298068624661161084365915969785419`20.000000000000007\
*^-20
There is no point applying N to a symbolic answer, it won't make any
difference. Both answers given by Mathematica (numerical and symbolic)
are perfectly correct, you just have to understand what they mean.
A
- References:
- summing 1/(n!) from 21 to Infinity
- From: Sampo Smolander <sampo.smolander+newsnspam@helsinki.fi>
- summing 1/(n!) from 21 to Infinity