values of ${(1+1/n)}^n$ for

The following table gives the numerator and denominator of ${\left(1+\frac{1}{n}\right)}^n$ as well as the decimal expansion to 20 places.

$n$	Numerator of ${\left(1+\frac{1}{n}\right)}^n$	Denominator of ${\left(1+\frac{1}{n}\right)}^n$	Decimal value of ${\left(1+\frac{1}{n}\right)}^n$
1	2	1	2.0000000000000000000
2	9	4	2.2500000000000000000
3	64	27	2.3703703703703703704
4	625	256	2.4414062500000000000
5	7776	3125	2.4883200000000000000
6	117649	46656	2.5216263717421124829
7	2097152	823543	2.5464996970407131139
8	43046721	16777216	2.5657845139503479004
9	1000000000	387420489	2.5811747917131971820
10	25937424601	10000000000	2.5937424601000000000
11	743008370688	285311670611	2.6041990118975308782
12	23298085122481	8916100448256	2.6130352902246781603
13	793714773254144	302875106592253	2.6206008878857322211
14	29192926025390625	11112006825558016	2.6271515563008693884
15	1152921504606846976	437893890380859375	2.6328787177279190470
16	48661191875666868481	18446744073709551616	2.6379284973665998588
17	2185911559738696531968	827240261886336764177	2.6424143751831096203
18	104127350297911241532841	39346408075296537575424	2.6464258210976854673
19	5242880000000000000000000	1978419655660313589123979	2.6500343266404449073
20	278218429446951548637196401	104857600000000000000000000	2.6532977051444201339
21	15519448971100888972574851072	5842587018385982521381124421	2.6562632139261049855
22	907846434775996175406740561329	341427877364219557396646723584	2.6589698585377882029
23	55572324035428505185378394701824	20880467999847912034355032910567	2.6614501186387814545
24	3552713678800500929355621337890625	1333735776850284124449081472843776	2.6637312580685940367
25	236773830007967588876795164938469376	88817841970012523233890533447265625	2.6658363314874199930

With a large enough value of $n$, this formula approximates the natural log base $e$. For example, with $n$ set to ten million, we get 2.7182816925449662712, which is 0.000000135914078964161737245574 short of 2.7182818284590452354 (this calculation took almost four minutes with Mathematica 4.2). It is less computationally intensive to use

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

which with $n$ set to 100 gives in less than a second a result to 20 places that is indistinguishable from $e$.

Title	values of ${(1+1/n)}^n$ for
Canonical name	ValuesOf11nnFor0N26
Date of creation	2013-03-22 17:02:26
Last modified on	2013-03-22 17:02:26
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	5
Author	PrimeFan (13766)
Entry type	Data Structure
Classification	msc 33B99