Big Oh question

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November 22nd 2011, 07:52 PM
aukie

Big Oh question
I am trying to prove an idientity I have come accross in a book:

I have managed to prove,

using various facts such as
- $O\left(O\left(f\left(z\right)\right)\right)=O\left (f\left(z\right)\right)$ ,
- $g\left(z\right)O\left(f\left(z\right)\right)=O(g(z )f(z))$
- and $\log z=O\left(z\right)$ .
But I can't prove the above more difficult identity. Can anyone help me with this?
Thanks for reading.
November 22nd 2011, 11:10 PM
CaptainBlack

Re: Big Oh question
Quote:
Originally Posted by aukie

I am trying to prove an idientity I have come accross in a book:

$\log\left(\log t-\log\log t+O\left(\frac{\log\log t}{\log t}\right)\right)=\log\log t-\frac{\log\log t}{\log t}-\frac{1}{2}\left(\frac{\log\log t}{\log t}\right)^{2}+O\left(\frac{\log\log t}{\left(\log t\right)^{2}}\right)$

I have managed to prove,

$\log\left(\log t+O\left(\log\log t\right)\right)=\log\log t+O\left(\frac{\log\log t}{\log t}\right)$

using various facts such as

$O\left(O\left(f\left(z\right)\right)\right)=O\left (f\left(z\right)\right)$ ,
$g\left(z\right)O\left(f\left(z\right)\right)=O(g(z )f(z))$
and $\log z=O\left(z\right)$ .

But I can't prove the above more difficult identity. Can anyone help me with this?
Thanks for reading.
I think you need to observe that for large enough then so:

and now expand the big log on the right as a power series.

CB
November 23rd 2011, 12:18 PM
aukie

Re: Big Oh question

This is precisely the part I was stuck on, was not sure how to deal with all terms that arise from the 3rd order expansion of the "big" log term on the right. But then I realized the $O(\frac{\log \log t}{(\log t)^2})$ term absorbs most of them.