How do we figure out a value for the expression 1+1/(1+1/(1+1/(1+1/(.... ? One way to get close is to write a short program to do it. We can apply the formula f(x) = (1+1/x) to itself over and over again. This is what the first couple of steps look like:

1 1
1+1/(1) 2
1+1/(1+1/(1)) 1.5
1+1/(1+1/(1+1/(1))) 1.6666666666667
1+1/(1+1/(1+1/(1+1/(1)))) 1.6

How about the expression sqrt(1+sqrt(1+sqrt(1+sqrt(1+.... ? We can do the same sort of program, except using the function f(x)=sqrt(1+x) instead.

1 1
sqrt(1+1) 1.4142135623731
sqrt(1+sqrt(1+1)) 1.55377397403
sqrt(1+sqrt(1+sqrt(1+1))) 1.5980531824786
sqrt(1+sqrt(1+sqrt(1+sqrt(1+1)))) 1.6118477541253

Now we can compare the two functions side by side. Are they the same? Do they get close to each other? Look at the results from the two programs run for a lot more steps below.

iteration f(x) = 1+1/x f(x) = sqrt(1+x)
0 1 1
1 2 1.4142135623731
2 1.5 1.55377397403
3 1.6666666666667 1.5980531824786
4 1.6 1.6118477541253
5 1.625 1.6161212065081
6 1.6153846153846 1.6174427985274
7 1.6190476190476 1.6178512906097
8 1.6176470588235 1.6179775309347
9 1.6181818181818 1.6180165422315
10 1.6179775280899 1.6180285974702
11 1.6180555555556 1.618032322752
12 1.618025751073 1.6180334739282
13 1.6180371352785 1.6180338296612
14 1.6180327868852 1.6180339395888
15 1.6180344478217 1.6180339735583
16 1.6180338134001 1.6180339840554
17 1.6180340557276 1.6180339872992
18 1.6180339631667 1.6180339883016
19 1.6180339985218 1.6180339886114
20 1.6180339850174 1.6180339887071
21 1.6180339901756 1.6180339887367
22 1.6180339882053 1.6180339887458
23 1.6180339889579 1.6180339887486
24 1.6180339886704 1.6180339887495
25 1.6180339887802 1.6180339887498
26 1.6180339887383 1.6180339887499
27 1.6180339887543 1.6180339887499
28 1.6180339887482 1.6180339887499
29 1.6180339887505 1.6180339887499
30 1.6180339887496 1.6180339887499
31 1.61803398875 1.6180339887499
32 1.6180339887499 1.6180339887499
33 1.6180339887499 1.6180339887499
34 1.6180339887499 1.6180339887499
35 1.6180339887499 1.6180339887499
36 1.6180339887499 1.6180339887499
37 1.6180339887499 1.6180339887499
38 1.6180339887499 1.6180339887499
39 1.6180339887499 1.6180339887499
40 1.6180339887499 1.6180339887499