[AussieKenDoll]

is it ΔV=±1 or what?
What is ΔV?

[Corribus]

This is something you can just look up in a textbook or on google.

[AussieKenDoll]

Yea I know that The mode in question must involve a changing dipole moment during the vibration. This can involve a bending process as well as a stretching process.
But can I show this statement mathematically as ΔV=±1  instead of that explanation?

[Corribus]

These are two completely different conditions - the requirement of a change in dipole is not the same as the requirement for a change in the quantum number of ± 1. Both requirements can be derived by evaluating the transition moment integral, but that's a fairly advanced mathematical exercise. The dipole change condition is easily shown by symmetry considerations.

You can see a little more here, although this link doesn't provide the actual derivation.

Note that the Δν = ±1 selection rule is only strictly valid in the harmonic approximation. In real systems that have anharmonic contributions, overtones with Δν = ±2, etc., become weakly allowed.