Using Newton's method to find $M$, the unique positive zero of $x^2+x-1$. Obtain a recursive formula for the error term $e_n$ use it to prove $a_n \rightarrow M$
Recursive formula for $a_n:\quad$ $a_{n+1} = a_n - \frac{a_n^2+a_n-1}{2a_n + 1} = \frac{2a_n^2 + a_n - a_n^2-a_n+1}{2a_n+1}= \frac{a_n^2+1}{2a_n+1}$
Recursive formula for $e_n:\quad$ $e_{n+1} = \frac{(e_n+M)^2+1}{2(e_n+M)+1}$
To show that $a_n \rightarrow M$, it is sufficient to show that for every $\epsilon > 0$ holds $|e_n| < \epsilon$.
How should I do that?