Approximating roots
Task number: 2877
Determine the limit of the recurrent sequence \(a_1= 1\) and \(\displaystyle a_{n+1}=\frac12\left(a_n+\frac c{a_n}\right)\), where \(c\) is a positive real number.
Using this result, calculate \(\sqrt 7\) to 4 decimal places.
Result
Using a method similar to the preceding example, we can show that \(\displaystyle \lim a_n=\sqrt c\).
For the given precision it suffices to compute \(a_5\) a \(c=7\).
We have \(a_1=1\), \(a_2=4\), \(a_3=\frac{23}{8}\), \(a_4=\frac{977}{368}\doteq 2{,}654\), \(a_5=\frac{1902497 }{719072}\doteq 2{,}6457670442 \), zatímco \(\sqrt 7 \doteq 2{,}64575131106459\).
Note that \(a_6=\frac{7238946623297}{2736064645568}\doteq 2{,}64575131111137 \), which agrees with \(\sqrt 7\) to nine decimal places.
