# Thread: Big O, Little O and Logarithm Integral

1. ## Big O, Little O and Logarithm Integral

Hello everyone,

I've spent the last hours trying to understand Big O and little O notations in depth. Usually the limit of the quotient method works but here, I'm dealing with integrals which seems to make things harder.

integral(dt/ln(t),t=2..sqrt(x))=O(sqrt(x)) when x->+∞
integral(dt/(ln(t))^2,t=sqrt(x)..x)=o(integral(dt/ln(t),t=sqrt(x)..x) when x->+∞

I'd really appreciate some tips or the general method on proving these two assertions. Sorry for not knowing Latex, I'll get to it soon ^^

Thanks for any help!

2. Originally Posted by DwProd

integral(dt/ln(t),t=2..sqrt(x))=O(sqrt(x)) when x->+∞
Put:

$I(x)=\displaystyle \int_{t=2}^{\sqrt{x}}\dfrac{1}{\ln(t)}\;dt$

Now $\ln (t)$ is a positive increasing function on $(1,\infty)$ so $1/\ln(t)$ is decreasing so:

$I(x)<\displaystyle \int_{t=2}^{\sqrt{x}}\dfrac{1}{\ln(2)}\;dt=\dfrac{ 1}{\ln(2)} \times (\sqrt{x}-2)$

and you should be able to complete the proof from there.

CB

3. Yes that is what I've been able to do so far but I'm not sure this allows me to verify the first assertion. I think it has to do with the
∃x0>0 ∃M>0 ∀x>x0 abs(f(x))<M*abs(g(x)) => f(x)=O(g(x)) but I'm not even sure it is a proper definition and why the condition x->+∞ is useful here. Thanks for your answer by the way!

4. Originally Posted by DwProd
integral(dt/(ln(t))^2,t=sqrt(x)..x)=o(integral(dt/ln(t),t=sqrt(x)..x) when x->+∞
Let:

$\displaystyle I(x)=\int_{\sqrt{x}}^x \dfrac{1}{(\ln(t))^2}\; dt$

again we observe that on $(1,\infty)$ $1/\ln(t)$ is decreasing so for $t>\sqrt{x}$:

$\dfrac{1}{(\ln(t))^2}<\dfrac{1}{\ln(\sqrt{x})}\tim es \dfrac{1}{\ln(t)}$

CB

5. Ok I think I get it, I can just create the quotients with those inequalities (since everything is positive and ≠ 0) and then just use the Squeeze theorem.
If this is all I need to do, please just confirm so that I can put the little [SOLVED] Thank you very much for your help CaptainBlack!

6. Originally Posted by DwProd
Ok I think I get it, I can just create the quotients with those inequalities (since everything is positive and ≠ 0) and then just use the Squeeze theorem.
If this is all I need to do, please just confirm so that I can put the little [SOLVED] Thank you very much for your help CaptainBlack!
I don't think you need the squeeze theorem just the definition of big-O and little-o notation

CB