Calculus Problem: Intgral Inequality w/ Positive Numbers

[sit.think.solve]

Suppose that
$$\alpha_1,...,\alpha_n$$
are positive numbers. Show that
$$\int_{1}^{\infty}...\int_{1}^{\infty}\frac{dx_1...dx_n}{{x_1}^{\alpha_1}+...+{x_n}^{\alpha_n}}<\infty$$
if
$$\frac{1}{\alpha_1}+...+\frac{1}{\alpha_n}<1$$

 

[daudaudaudau]

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

 

[Mark44]

Suppose that
$$\alpha_1,...,\alpha_n$$
are positive numbers. Show that
$$\int_{1}^{\infty}...\int_{1}^{\infty}\frac{dx_1...dx_n}{{x_1}^{\alpha_1}+...+{x_n}^{\alpha_n}}<\infty$$
if
$$\frac{1}{\alpha_1}+...+\frac{1}{\alpha_n}<1$$

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

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.