Comments on A095822.

The rational numbers r(n):= sum(1/k!,k=0..n) + 1/(n*n!), n>=1  (written in lowest 
terms, i.e. without common factors >1 in the numerator and denominator) are, for 1 .. 15:

[3, 11/4, 49/18, 87/32, 1631/600, 11743/4320, 31967/11760, 876809/322560, 8877691/3265920, 

4697191/1728000, 1193556233/439084800, 2232105163/821145600, 2222710781/817689600, 

3317652307271/1220496076800, 53319412081141/19615115520000]


Their values (maple9 with 20 digits) are

[3., 2.7500000000000000000, 2.7222222222222222222, 2.7187500000000000000, 2.7183333333333333333, 

2.7182870370370370370, 2.7182823129251700680, 2.7182818700396825397, 2.7182818317656280619, 

2.7182818287037037037, 2.7182818284759572638, 2.7182818284601415388, 2.7182818284591121130, 

2.7182818284590490869, 2.7182818284590454454]

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Euler's number e:=sum(1/k!,k=0..infty)= exp(1) has as upper bound r(n), for every n>=1.

The errors 0<=  r(n)- exp(1)  are (maple9, 20 digits), for n=1..15:
      
[.2817181715409547646, 0.317181715409547646e-1, 0.39403937631769868e-2, 0.4681715409547646e-3,

 0.515048742880979e-4, 0.52085779918016e-5, 0.4844661248326e-6, 0.415806373043e-7, 0.33065828265e-8, 

0.2446584683e-9, 0.169120284e-10, 0.10963034e-11, 0.668776e-13, 0.38515e-14, 0.2100e-15]

Here e-k at the end of a decimal means 10^{-k}. 

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Note that exp(1): = sum(1/k!,k=0..infty) =  limit((1_1/n)^n,n to infty) =: e.

For a proof of these upper bounds see, .e.g., the Barner-Flohr Analysis I book quoted under this A-number.

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