values of ${\displaystyle \sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{i=0}^n\frac{1}{i!}}$ for

The following table gives the numerator and denominator of

$$\sum _{i=0}^n\frac{1}{i!}$$

as well as the decimal expansion to 20 places.

$n$	Numerator of ${\displaystyle \sum _{i=0}^n}{\displaystyle \frac{1}{i!}}$	Denominator of ${\displaystyle \sum _{i=0}^n}{\displaystyle \frac{1}{i!}}$	Decimal value of ${\displaystyle \sum _{i=0}^n}{\displaystyle \frac{1}{i!}}$
1	2	1	2.0000000000000000000
2	5	2	2.5000000000000000000
3	8	3	2.6666666666666666667
4	65	24	2.7083333333333333333
5	163	60	2.7166666666666666667
6	1957	720	2.7180555555555555556
7	685	252	2.7182539682539682540
8	109601	40320	2.7182787698412698413
9	98641	36288	2.7182815255731922399
10	9864101	3628800	2.7182818011463844797
11	13563139	4989600	2.7182818261984928652
12	260412269	95800320	2.7182818282861685639
13	8463398743	3113510400	2.7182818284467590023
14	47395032961	17435658240	2.7182818284582297479
15	888656868019	326918592000	2.7182818284589944643
16	56874039553217	20922789888000	2.7182818284590422591
17	7437374403113	2736057139200	2.7182818284590450705
18	17403456103284421	6402373705728000	2.7182818284590452267
19	82666416490601	30411275102208	2.7182818284590452349
20	6613313319248080001	2432902008176640000	2.7182818284590452353
21	69439789852104840011	25545471085854720000	2.7182818284590452354
22	611070150698522592097	224800145555521536000	2.7182818284590452354
23	1351405140967886501753	497154168055480320000	2.7182818284590452354
24	337310723185584470837549	124089680346647887872000	2.7182818284590452354
25	85351903640077042215979	31399210614030336000000	2.7182818284590452354

As the table shows, small values of $n$, are sufficient to give a good approximation of the natural log base $e$, which at $n=21$ is at 20 places undistinguishable from $e$.

Title	values of ${\displaystyle \sum \mathrm{@}\mathrm{@}\mathrm{@}{\mathrm{@}}_{i=0}^n\frac{1}{i!}}$ for
Canonical name	ValuesOfdisplaystylesumi0nfrac1iFor0N26
Date of creation	2013-03-22 17:06:13
Last modified on	2013-03-22 17:06:13
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Example
Classification	msc 33B99