Problem #PRU-61302

Problem

Old calculator I.

a) Suppose that we want to find $\sqrt[3]{x}$ ($x>0$) on a calculator that can find $\sqrt{x}$ in addition to four ordinary arithmetic operations. Consider the following algorithm. A sequence of numbers $\{y_n\}$ is constructed, in which $y_0$ is an arbitrary positive number, for example, $y_0=\sqrt{\sqrt{x}}$, and the remaining elements are defined by $y_{n+1}=\sqrt{\sqrt{x{y}_n}}$ ($n\ge 0$).

Prove that $\underset{n\to \mathrm{\infty }}{lim}{y}_n=\sqrt[3]{x}$.

b) Construct a similar algorithm to calculate the fifth root.