# Showing a sequence tends to infinity

• September 22nd 2009, 04:31 PM
cgiulz
Showing a sequence tends to infinity
Define a sequence recursively by $a_{n+1} = 2a_{n}^2$.

Given $a_{0} > 1/2$, show lim $a_{n} = \infty$.

Note: It can easily be shown that $a_{n} > 1/2$ so just assume this is given.
• September 23rd 2009, 12:32 AM
paweld
Let $b_n=2 a_n$.
We have the following reccurence for sequence $b_n$:
$
b_{n+1}=b_n^2
$

But for this sequence explicit formula can be easily obtained:
$
b_n=b_0^{2^n}
$

Because $b_0 > 1$ the sequence is not convergent.
• September 23rd 2009, 06:09 PM
cgiulz
Hey, sorry for the late response, thanks though.

So I can just assume $b_{n} > 1/2$ and just say since,

$b_{n} < a_{n}$ for all $n$?