Good morning,
I would love your help with this:
Given a sequence {$ a_{n} $} with this initial data:
$a_{2k}= \frac{a_{2k-1}}{2}$
$a_{2k+1}= a_{2k}+\frac{1}{2}$
I need to find $\liminf a_{n}$ and $\limsup a_{n}$
Thank you
Good morning,
I would love your help with this:
Given a sequence {$ a_{n} $} with this initial data:
$a_{2k}= \frac{a_{2k-1}}{2}$
$a_{2k+1}= a_{2k}+\frac{1}{2}$
I need to find $\liminf a_{n}$ and $\limsup a_{n}$
Thank you
If you denote $b_k=a_{2k+1}$ then you can show $b_k=\frac{b_{k-1}}2+\frac12$, which is equivalent to $b_k-1=\frac{b_{k-1}-1}2$. Hence the sequence $c_k=b_k-1$ is a geometric progression with quotient 1/2. This shows that
$\lim_{k\to\infty} a_{2k+1} = 1+\lim_{k\to\infty} c_k = 1.$
Almost in the same way you can show that $a_{2k}$ converges to 1/2. (Or you can use that you already know the limit of $a_{2k+1}$).