Question

Improper integrals and the P-test: can an integral of the following form be immediately evaluated by the P-test, if at all?

Answer 1

\[\int\limits_{4}^{\infty}\frac{ dx }{ x ^{P} }\]

Answer 2

The P-test states that an integral \[\int\limits_{1}^{\infty}\frac{ dx }{ x^{P} }\]
-diverges if P<1
-diverges if P=1
-is -1/(1-P) if P>1

Answer 3

\[\int_1^\infty\frac{dx}{x^p}=\int_1^4\frac{dx}{x^p}+\int_4^\infty\frac{dx}{x^p}\]

Answer 4

But if the integral from 1 to infinity is greater than the integral from four to infinity, then how do we know that the integral from 4 to infinity diverges?

Answer 5

If the integral over [1,infinity) diverges, then so will the integral over [4,infinity). That's because the intermediate integral over [1,4] is finite (assuming it also converges, but it will if it's in the form dx/(x^p).

Answer 6

Ohh, I see... So it'd be like adding a finite number to infinity.