Dispersion of the wave packet over time

[abeer-0101]

TL;DR

I am confused about how the taylor expansion controls the shape of the wave packet

since, in order to view the shape changes in our wave packet we are presented with the taylor expansion of the frequency
ω(k) = ω(k0) + (k − k0)dω/dk + 1/2*(k − k0)^2 (d^2ω/dk^2)
we are told that only the third term that is the
1/2*(k − k0)^2 (d^2ω/dk^2)
contributes to change in shape of the wave function over time.why is that?

 

[anuttarasammyak]

If there is no terms of second order and more
ω(k) = ω(k0) + (k − k0)dω/dk
\frac{d\omega}{dk}=\frac{\omega(k)-\omega(k0)}{k-k0}=v
constant. It means all the different k components have same speed so there happens no change in shape of the wave.