Calculus Problem: Intgral Inequality w/ Positive Numbers

[sit.think.solve]

Suppose that
<br /> \alpha_1,...,\alpha_n<br />
are positive numbers. Show that
<br /> \int_{1}^{\infty}...\int_{1}^{\infty}\frac{dx_1...dx_n}{{x_1}^{\alpha_1}+...+{x_n}^{\alpha_n}}&lt;\infty<br />
if
<br /> \frac{1}{\alpha_1}+...+\frac{1}{\alpha_n}&lt;1<br />

 

[daudaudaudau]

Hi. I've been thinking about this one, but I can't solve it. Where did you get this problem?

 

[Mark44]

Suppose that
<br /> \alpha_1,...,\alpha_n<br />
are positive numbers. Show that
<br /> \int_{1}^{\infty}...\int_{1}^{\infty}\frac{dx_1...dx_n}{{x_1}^{\alpha_1}+...+{x_n}^{\alpha_n}}&lt;\infty<br />
if
<br /> \frac{1}{\alpha_1}+...+\frac{1}{\alpha_n}&lt;1<br />

I would try breaking this down into smaller, easier problems. What do you get for this integral?
<br /> \int_{1}^{\infty}\frac{dx_1}{{x_1}^{\alpha_1}+...+{x_n}^{\alpha_n}}<br />

Note that this is an improper integral, so the limits will need to be 1 and, say, b, and you'll need to take the limit as b -> \infty.

For the integral to converge, are there any restrictions on \alpha_1?

Then work with the double integral, with dx_1 and dx_2. For this integral to converge, what restrictions must be placed on \alpha_1 and \alpha_2?

That's how I would tackle this.