3
votes
2answers
95 views

If a recursive sequence converges, must its inverse be divergent?

Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = ...
0
votes
1answer
34 views

Iterated Functions - designing iterator to converge to constant value

I came across an interesting iterated function: $$ x_n = \frac{x_{n-1}}{x_{n-1} + b} $$ This is an extremely simple example and it converges to the constant $1-b$. Can someone provide some insight to ...
3
votes
4answers
137 views

Proving convergence of a sequence

Let the following recursively defined sequence: $a_{n+1}=\frac{1}{2} a_n +2,$ $a_1=\dfrac{1}{2}$. Prove that $a_n$ converges to 4 by subtracting 4 from both sides. When I do that, I get: ...
0
votes
0answers
102 views

Probably easy recursion formula solving $x^x=a$

Let $a,x_0\in\mathbb{C}$ and set $$ x_{n+1}=\sqrt{x_n\cdot \sqrt[x_n]{a}}$$ For which $a$ and $x_0$ does $x=\lim\limits_{n\rightarrow\infty} x_n$ exist with $$x^x=a$$ Approximating it with a computer ...