Showing a sequence tends to infinity

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Sep 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.
Sep 23rd 2009, 12:32 AM
paweld

Let $b_n=2 a_n$ .
We have the following reccurence for sequence $b_n$ :
$<br /> b_{n+1}=b_n^2<br />$
But for this sequence explicit formula can be easily obtained:
$<br /> b_n=b_0^{2^n}<br />$
Because $b_0 > 1$ the sequence is not convergent.
Sep 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 $n$ ?

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