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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

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up vote 1 down vote accepted

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 $a_{2k+1}$).

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