Math and science::Analysis::Tao::06. Limits of sequences

Square root, expressed as the limit of a sequence

Let $c \in R$ be a real. The sequence defined recursively below converges, and it converges to $\sqrt{c}$ .

Let $x_{1} = c$ , and let $x_{n}$ be defined like so:

x_{n + 1} = \frac{1}{2} (x_{n} + \frac{c}{x_{n}})

Proof by induction (any other typical methods) can be used to show that sequences defined recursively like this do (or don't) converge.

Intuition

The outer $\frac{1}{2}$ factor can be seen as taking an average of the two inner terms. Inspecting the second term, $\frac{1}{2} (c + \frac{c}{c}) = \frac{c + 1}{2}$ , the third, $\frac{1}{2} (\frac{c + 1}{2} + c \frac{2}{c + 1})$ , and so on, one can be convinced that this sequences is increasing starting from the third term. In addition, it can be seen that the terms don't surpass $\sqrt{c}$ . While not a exactly convincing as a proof, you can check if a term did surpass $\sqrt{c}$ , then the following term would be less than that term.

Source

Abbott, p54

25.03.2021