Question

How do you prove that the 4-sd approximation to the value of `log_2(2+1/log_2(2+1/log_2(2+...)))` is 1.428?

Answer 1

`approx1.432`

Explanation:

Calling

`y = log_2(2+1/log_2(2+1/log_2(2+...)))` we have

`y = log_2(2+1/y)`

Now, calling

`y_{k+1} = log_2(2+1/y_k)`

substituting

`y_1 = 1.428` we obtain
`y_2=1.433` and sucessively
`y_3 =1.432` etc.

converging to

`( (y_1 = 1.428000000000000000000000000000), (y_2 = 1.433109072200861922041781326700), (y_3 = 1.431774625761248698196936857130), (y_4 = 1.432122371909281044466411003668), (y_5 = 1.432031697667491831893471498920), ( cdots), (y_21 = 1.432050448448111801269533316372), (y_22 = 1.432050448448122681455174642906), (y_23 = 1.432050448448119794875310617499), (y_24 = 1.432050448448120683053730317624), (y_25 = 1.432050448448120238964520467562), (y_26 = 1.432050448448120238964520467562) )`