On the convergence of sequences by recurrence

Question

Hello I'm studying some problems on computer science in which arises the following recurrence sequences:

$$x_{n+1} = \frac{1}{k + x_{n}}$$ $$x_{n+1} = \frac{1}{1/n + x_{n}}$$

$x_{0} = \epsilon >0$ for both and $k>0$.

What do you think about convergence on both?

You could try to think about how $x_n$ and $n$ compare as n grows. And then ask yourself if the sequence is monotone at some point. — Frostic, Apr 19 '18 at 20:29
For the 1st one you can try Banach fixed point theorem, here is an example — rtybase, Apr 19 '18 at 21:43

score 0 · Answer 1 · answered Apr 19 '18 at 20:38

The first one converges for $k > .6$.

From $x_{n+1} = \frac{1}{k + x_{n}}$, $x_n < \frac1{k}$.

Therefore $x_n > \frac1{k+1/k}$.

Therefore $x_{n+2}-x_{n+1} = \frac{1}{k + x_{n+1}}-\frac{1}{k + x_{n}} = \frac{x_n-x_{n+1}}{(k + x_{n+1})(k + x_{n})} $ so $|\dfrac{x_{n+2}-x_{n+1}}{x_n-x_{n+1}} | = |\frac{1}{(k + x_{n+1})(k + x_{n})}| \lt |\frac{1}{(k + 1/(k+1/k))^2}| = |\frac{1}{(k + k/(k^2+1))^2}| = |\frac{(k^2+1)^2}{(k(k^2+1) + k)^2}| = |\frac{(k^2+1)^2}{(k^3+2k)^2}| \lt 1 $ for $x > .6$ according to Wolfy.

On the convergence of sequences by recurrence

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