- #1

crazedbeat

- 19

- 0

**Determine divergence/convergence of an integral??**

Given that

[tex]F(x) = \int_{0}^{x} f(s) ds[/tex]

and

[tex]F(\infty)=1[/tex]

prove that for any

[tex]\alpha \geq 1[/tex],

[tex]\int_{0}^{\infty} x^{\alpha} dF(x) = \alpha \int_{0}^{\infty} x^{\alpha - 1}(1 - F(x))dx[/tex]

where two sides either converge or diverge together.

note: some of it didn't come out correctly. on the left hand side it means x raised alpha, and on the left x raised to (alpha - 1).

I have NO idea how to even START the problem. I've been doing convergence/divergence by finding limits, ratio test, etc. But these are not even close-- these are integrals and they don't make any sense. Any guidance into what i should be looking at would be greatly appreciated.

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