How are the Taylor series taken?

A lecture slide from numerical analysis shows how the taylor series is taken.

$$frac{u_{i+1}-u_{i}}{h}=u_{i}^{prime}+frac{h}{2} u_{i}^{prime prime}+frac{h^{2}}{6} u_{i}^{prime prime prime}+mathcal{O}left(h^{3}right)$$

I normally do not have trouble with the Taylor series, but I cannot see how one goes from the left hand side to the right hand side.

Due to covid-19 I cannot go to uni and ask my professor. Hopefully someone could provide me with the insight to solve this.

Ter