Given a sequence as below $$ \left\{ \begin{array} {l} x_1=1\\ x_{n+1}=3-\frac{x_n+2}{2^{x_n}} \end{array} \right. $$ Prove that the sequence converges to a limit, and find the limit?

  • $\begingroup$ What have you tried so far? Have you already a guess for the limit? Hint: Recurrence relations are often solved via induction (on two steps). $\endgroup$ – BIS HD Nov 12 '13 at 13:24

Hint: The function $$f(x) = (3-x)2^x - (x+2)$$ is positive on $[0,2)$ and $f(2)=0$.

The function $g(x) = x+2-2^x$ is positive on $[0,2)$ and $g(2)=0$.

This makes the sequence $(a_n)_{n\geq 1}$ increasing and upper bounded by $2$. Its limit $\ell$ therefore lies in $(1,2]$, and satisfies $f(\ell) = 0$, so $\ell = 2$


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