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[parent] values of $\displaystyle \sum_{i = 0}^n \frac{1}{i!}$ for $0 < n < 26$ (Example)

The following table gives the numerator and denominator of

$\displaystyle \sum_{i = 0}^n \frac{1}{i!}$
as well as the decimal expansion to 20 places.
$ n$ Numerator of $ \displaystyle \sum_{i = 0}^n \frac{1}{i!}$ Denominator of $ \displaystyle \sum_{i = 0}^n \frac{1}{i!}$ Decimal value of $ \displaystyle \sum_{i = 0}^n \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$.



"values of $\displaystyle \sum_{i = 0}^n \frac{1}{i!}$ for $0 < n < 26$" is owned by PrimeFan.
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Cross-references: natural log base, approximation, sufficient, places, decimal expansion, denominator, numerator

This is version 2 of values of $\displaystyle \sum_{i = 0}^n \frac{1}{i!}$ for $0 < n < 26$, born on 2007-05-19, modified 2007-06-12.
Object id is 9401, canonical name is ValuesOfDisplaystyleSum_i0nFrac1iFor0N26.
Accessed 433 times total.

Classification:
AMS MSC: 33B99 (Special functions :: Elementary classical functions :: Miscellaneous)
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