## Approximating roots

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.