Spring hanging from ceiling

I have a question that's being bugging me around. This might be simple but I can't figure it out. If there's a spring hanging from the ceiling and we want to prove that there's a Simple Harmonic Oscilation then why don't we account for the gravitational force?

The sum of forces equals to mass times aceleration (ma) according to newton's 2nd law, so the sum of forces is the gravitional force plus the force by the spring (there should be vectors above the forces of course).

This is a differential equation since aceleration equals the second derivitive of position and to prove that there is a S.H.O. the equation must be
(d²x)/(dt²) + (K/m)x = 0

But in fact what I actually get is
(d²x)/(dt²) + (K/m)x + g = 0

I know that in this equation g is a constant but my question is is g relevant if we want to prove that there is a S.H.O?

And following that could anybody tell me how do we demonstrate that the period is in fact
T=2π√(m/K)

Thanks for your time