Weird but wonderful

Jacob Bernouilli

Most people love \pi. But I love e more.
You might know that e\approx 2.71828182846. What I find fascinating about e is not only the omnipresence of the number (it’s everywhere!) but also how slow it progresses to its approximate value. While \pi comes out fairly straight at 3.1415926... after only a moderate number of iterations, e takes a lot longer to get there.

You can test it yourself with this short python program I wrote. You can download Python for free. Just plug it in and run:

n=0
for n in range(start,stop,step):
     e=((1+1/n)**n)
     print(n,e)

Of course you have to substitute start with your start value, say 1, stop with your end value, say 100 and step with the steps you want to take, say 10.

Here you have the first 50 values for n:

1 2.0
2 2.25
3 2.37037037037037
4 2.44140625
5 2.4883199999999994
6 2.5216263717421135
7 2.546499697040712
8 2.565784513950348
9 2.5811747917131984
10 2.5937424601000023
11 2.6041990118975287
12 2.613035290224676
13 2.6206008878857308
14 2.6271515563008685
15 2.6328787177279187
16 2.6379284973666
17 2.64241437518311
18 2.6464258210976865
19 2.650034326640442
20 2.653297705144422
21 2.656263213926108
22 2.658969858537786
23 2.6614501186387796
24 2.663731258068599
25 2.665836331487422
26 2.6677849665337465
27 2.6695939778125704
28 2.6712778534408463
29 2.6728491439808066
30 2.6743187758703026
31 2.6756963059146854
32 2.676990129378183
33 2.678207651253779
34 2.6793554280957674
35 2.6804392861534603
36 2.6814644203008586
37 2.6824354773085255
38 2.6833566262745787
39 2.6842316184670922
40 2.685063838389963
41 2.6858563475377526
42 2.686611922032571
43 2.687333085118294
44 2.6880221353133043
45 2.688681170884324
46 2.689312111189782
47 2.6899167153502597
48 2.6904965986289264
49 2.691053246842418
50 2.691588029073608

But if you raise the stakes, say to n=10^9, it’s still only 2.7182820532347876.... Just 5 decimals right.

Worse, it was more accurate at lower numbers, like at n=10,000,000, namely 2.7182816941320818... with 6 decimals right.
Conclusion: e is weird, and thus wonderful.

Leonard Euler