Limits are plugged in terms become infinite

[DylanB]

Homework Statement

find the integral from 0 to infinity of t^(-1)*e^(-st) dt

Homework Equations

e^x series expansion

The Attempt at a Solution

I expanded e^(-st) into its series and then integrated, it appears to diverge since when the limits are plugged in terms become infinite, but I am unsure since the series alternates with (-1)^n.

 

[Dick]

Sure it diverges. exp(-st) approaches 1 as t->0. So near 0 the integral diverges like 1/t. If s=0 it's exactly 1/t.

 

[DylanB]

Is showing that the function has a diverging asymptote at one of the endpoints strong enough proof to say that it's integral diverges to +inf?

 

[Dick]

Sure. More formally write a comparison test for the integral on the interval of say t from 0 to 1/s. If it's greater than something that diverges, then it diverges.