Basic Analysis I — Real Analysis · Problem 2.2.11

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · recursively-defined-sequences

Problem

Let r>0r > 0. Show that starting with an arbitrary x1≠0x_1 \neq 0, the sequence defined by xn+1:=xn−xn2−r2xnx_{n+1} := x_n - \frac{x_n^2 - r}{2x_n} converges to r\sqrt{r} if x1>0x_1 > 0 and −r-\sqrt{r} if x1<0x_1 < 0.

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