# 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?

Jul 28, 2016

$\approx 1.432$

#### Explanation:

Calling

$y = {\log}_{2} \left(2 + \frac{1}{\log} _ 2 \left(2 + \frac{1}{\log} _ 2 \left(2 + \ldots\right)\right)\right)$ we have

$y = {\log}_{2} \left(2 + \frac{1}{y}\right)$

Now, calling

${y}_{k + 1} = {\log}_{2} \left(2 + \frac{1}{y} _ k\right)$

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) )