Mathematica erroneously says this integral does not converge

The integral does not converge if a and b are real and have opposite signs:

Series[(ArcTan[a*x] - ArcTan[b*x])/x, {x, Infinity, 2}]

As @shadowray pointed out, it does converge if a and b are positive and Integrate[] is passed that as an assumption. It also works with

Integrate[(ArcTan[a*x] - ArcTan[b*x])/x, {x, 0, Infinity}, Assumptions -> a < 0 && b < 0]

Although Integrate usually finds the condition in which the integral converges,

Integrate[Exp[-a x], {x, 0, Infinity}]

it's better to provide your own condition in the form of Assumptions.

Integrate[(ArcTan[a*x] - ArcTan[b*x])/x, {x, 0, Infinity}, Assumptions->{Re[a]>0, Re[b]>0}]

Integrate[(ArcTan[a*x] - ArcTan[b*x])/x, {x, 0, Infinity}, Assumptions->{Re[a]<0, Re[b]<0}]