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

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 $\displaystyle\sum_{i=0}^{n}\frac{1}{i!}$ 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 $e$.

Title values of $\displaystyle\sum@\slimits@@@_{i=0}^{n}\frac{1}{i!}$ for $0 ValuesOfdisplaystylesumi0nfrac1iFor0N26 2013-03-22 17:06:13 2013-03-22 17:06:13 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Example msc 33B99