Convergence Problem... Problems

**scg780** · April 21st 2010, 07:00 PM

Currently taking integral calculus and I have a problem that I have no idea how to approach.

Given the integral from 1 to infinity of dx/(x^p *(ln(x))^a)

- For what values of p and a does this converge

Even if you cant get the full solution, any thoughts on how to approach it would be greatly appreciated.

**CaptainBlack** · April 22nd 2010, 12:19 AM

Originally Posted by scg780

Currently taking integral calculus and I have a problem that I have no idea how to approach.

Given the integral from 1 to infinity of dx/(x^p *(ln(x))^a)

- For what values of p and a does this converge

Even if you cant get the full solution, any thoughts on how to approach it would be greatly appreciated.

For this integral to converge requires that the limit:

$\lim_{u\to \infty} \int_1^u\frac{1}{x^p (\ln(x))^a}\;dx$

exists. Now as $\ln(x)$ increases more slowly than any power of $x$ this is equivalent to asking for what values of $p$ does:

$\lim_{u\to \infty} \int_1^u\frac{1}{x^p}\;dx$

converge (with $p=1$ possibly being a special case).

CB

**scg780** · April 22nd 2010, 06:44 AM

ok. Thank you. but i'm still confused on what the value of "a" would be

**chisigma** · April 22nd 2010, 12:46 PM

Integrating by parts we obtain...

$\int \frac{dx}{x^{p}\cdot \ln^{a} x} = \frac{1}{1-p} \cdot \frac{1}{x^{p-1}\cdot \ln^{a} x} + \frac{a}{1-p} \int\frac{dx}{x^{p}\cdot \ln ^{a+1} x}$ (1)

... and then with another integration by parts...

$\int \frac{dx}{x^{p}\cdot \ln^{a} x} = \frac{1}{1-p} \cdot \frac{1}{x^{p-1}\cdot \ln^{a} x} + \frac{a}{(1-p)^{2}} \cdot \frac{1}{x^{p-1}\cdot \ln^{a+1} x} +$

$+ \frac{a\cdot (a+1)}{(1-p)^{2}}\cdot \int \frac{dx}{x^{p}\cdot \ln^{a+2} x}$ (2)

If now we proceed indefinitely we obtain...

$\int \frac{dx}{x^{p}\cdot \ln^{a} x} = \frac{1}{1-p} \cdot \frac{1}{x^{p-1}\cdot \ln^{a} x} + \frac{a}{(1-p)^{2}} \cdot \frac{1}{x^{p-1}\cdot \ln^{a+1} x} + \dots$

$\dots + \frac{a\cdot (a+1) \dots (a+n-1)}{(1-p)^{n}}\cdot \frac{1}{x^{p-1}\cdot \ln^{a+n} x} + \dots$ (3)

At this point a careful investigation of (3) permits us to conclude that the [improper] integral...

$\int_{1}^{\infty} \frac{dx}{x^{p}\cdot \ln^{a} x}$ (4)

... converges if...

a) $p>1$

b) $a$ is a non positive integer...

If $a= -k, k\ge 0$ is ...

$\int_{1}^{\infty}\frac{\ln^{k} x}{x^{p}} = \frac{k!}{(p-1)^{k+1}}$ (5)

The convergence in the case $a<0$ but non integer can be easily verified by comparison...

Kind regards

$\chi$ $\sigma$

Thread: Convergence Problem... Problems

LinkBack

Thread Tools

Display

Convergence Problem... Problems

Similar Math Help Forum Discussions

L^2 convergence problems.

convergence problems

Two problems regarding convergence

Convergence problems

Help in Interval of convergence problems!

Search Tags