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Homevalues of $(1 + 1/n)^n$ for $0 < n < 26$

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# values of $(1+1/n)^{n}$ for $0<n<26$

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

$n$ | Numerator of $\left(1+{1\over n}\right)^{n}$ | Denominator of $\left(1+{1\over n}\right)^{n}$ | Decimal value of $\left(1+{1\over n}\right)^{n}$ |

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 $e$.

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## Comments

## usefulness of this entry...

/me wonders how long will take before someone questions it.

Like on http://planetmath.org/encyclopedia/TableOfValuesForSinx.html

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: usefulness of this entry...

Drini said: me wonders how long will take before someone questions it.

I think you are already questioning the usefulness... and for a good reason. Perhaps, I can see some remote pedagogical value on showing the decimal expansion of consecutive values of (1+1/n)^n, just so one can understand the rate of convergence towards e ... but what is the point of showing the numerator and denominator???

The numerator is clearly (n+1)^n and the denominator is n^n, do we really need tables for those?

T

## Re: usefulness of this entry...

It's of no use for show-offs, that's for sure. But it is useful for people, like me, who go for a sanity-check every once in a while. Like, I had a calculator once that had the fourth digit cell from the right its bottom left segment blown out. If I hadn't thought to try inputting a bunch of 8s, I might have used it for something important and paid $1,000 too much for something (or $100.0, but still...

Before I start rambling, let me just say I find this entry very useful. It's one thing to know that such and such formula approximates a certain constant, but it feels more real if you have some awareness of what kind of values give better approximations, or what kind of formulas. Your table of sine values is useful for checking your calculator's working as you expect (you might not have any segments blown out, but you might've forgotten you've got it set to gradians or something).

## Re: usefulness of this entry...

I was questioning the usefulness of usefulness questioning.

f

G -----> H G

p \ /_ ----- ~ f(G)

\ / f ker f

G/ker f

## Re: usefulness of this entry...

> I was questioning the usefulness of usefulness questioning.

How useless!