# Integral limit

• July 12th 2011, 04:22 AM
durrrrrrrr
Integral limit
Does the integral in the attached image converge?

Can you give me a hint how to start solving this question. Thanks
• July 12th 2011, 11:36 AM
Opalg
Re: Integral limit
Here's the informal argument. I'll leave it to you to make it rigorous.

The singularity occurs at t=1, so we need to think about what the function $(\log|\log t|)^7$ looks like when t is close to 1.

For t close to 1, $|\log t|$ is approximately 1–t, so we should look at the function $\bigl(\log(1-t)\bigr)^7.$ In fact, that function is negative for 0<t<1, so it would be more convenient to look at its negative, namely $\bigl|\log(1-t)\bigr|^7.$

As x decreases to 0, |log x| increases to infinity, but more slowly than any negative power of x: $|\log x|\leqslant x^{-\alpha}$ for any $\alpha>0.$ Put x = 1–t to get $\bigl|\log(1-t)\bigr|^7 \leqslant (1-t)^{-7\alpha}.$ Choosing $\alpha = 1/14$, you see that $\bigl|\log(1-t)\bigr|^7 \leqslant (1-t)^{-1/2}.$ The integral $\int_{1/2}^1(1-t)^{-1/2}dt$ converges, hence (by the comparison test for improper integrals) so should the integral that you started with.
• July 13th 2011, 12:43 AM
durrrrrrrr
Re: Integral limit
Quote:

Originally Posted by Opalg
For t close to 1, $|\log t|$ is approximately 1–t

How do I know this is true?

Quote:

Originally Posted by Opalg
As x decreases to 0, |log x| increases to infinity, but more slowly than any negative power of x: $|\log x|\leqslant x^{-\alpha}$ for any $\alpha>0.$

How do I prove this?

Thanks
• July 13th 2011, 01:27 AM
FernandoRevilla
Re: Integral limit
Verify $\lim_{t\to 1^-}\frac{|\log t|}{1-t}=\ldots=1$ and $\lim_{x\to 0^+}\frac{|\log x|}{x^{-\alpha}}=\ldots=0\;\;(\alpha>0)$
• July 13th 2011, 01:40 AM
durrrrrrrr
Re: Integral limit
Quote:

Originally Posted by FernandoRevilla
Verify $\lim_{t\to 1^-}\frac{|\log t|}{1-t}=\ldots=1$

Thank you. I managed to prove the second limit, but I don't know how to prove this one.

Quote:

Originally Posted by Opalg
The integral $\int_{1/2}^1(1-t)^{-1/2}dt$ converges

And I how do I prove this?
• July 13th 2011, 01:47 AM
FernandoRevilla
Re: Integral limit
Use the substitution $u=1-t$ and the well known property: $\int_{0}^1\frac{du}{u^p}$ converges iff $p<1$ .