Hi,
Let a(n) be a sequence such that lim(a(2n)-a(n))=0.
Prove or disprove:the sequence a(n) converges.
Thank's in advance.
Let $\displaystyle n = \prod_{p\text{ is prime}} p^{b_p}$ be the prime factorization of $n$.
Then let
$\displaystyle a_n = \prod_{\begin{matrix} p\text{ is prime} \\ p \neq 2\end{matrix}}p^{b_p}$
So, here is a quick table of values so you can see the pattern:
$\displaystyle \begin{matrix}n & a_n \\ \hline 1 & 1 \\ 2 & 1 \\ 3 & 3 \\ 4 & 1 \\ 5 & 5 \\ 6 & 3 \\ 7 & 7 \\ 8 & 1 \\ 9 & 9 \\ 10 & 5 \\ 11 & 11 \\ 12 & 3\end{matrix}$
Here it is easy to see that for all $n$, $a_{2n} = a_n$. But, this sequence quite clearly does not converge.