Problem 84E

84. (a) Let $a_{1}=a, a_{2}=f(a), a_{3}=f\left(a_{2}\right)=f(f(a)), \ldots$, $a_{n+1}=f\left(a_{n}\right)$, where $f$ is a continuous function. If $\lim _{n \rightarrow \infty} a_{n}=L$, show that $f(L)=L$

(b) Illustrate part (a) by taking $f(x)=\cos x, a=1$, and estimating the value of $L$ to five decimal places.

Step-by-Step Solution