I was studying a numerical solution to the heat equation, specifically this one. I will summarise it at the end of this question so you don't need to click the link. I already asked a different question about it here. However that was about stability, and this is not, so I am asking a new question rather than editing.
Question: I need to compare my answer with the theoretical order of accuracy of this method, which is not mentioned in this Wikipedia page, and I don't know what it is at all. What is it, how do I calculate it?
Edit: Is it the $O(k)$ and $O(h^2)$ terms after the approximations for the derivatives? If so the problem still remains that I don't know how to calculate these of compare them with anything.
Edit 2: From the comments, I know what $O$ notation is. But how can it be manipulated in the step from equation $(2)$ to $(3)$? I don't understand how addition or subtraction of these two factors could work, and don't see how we can move things around to achieve division.
PS: I don't know quite what the (order) tag means, if it is appropriate then please edit it in. Thanks in advance for any help :)
Summary of Wikipedia page:
We have the differential equation $$U_t=U_{xx}\tag1$$ We want to numerically approximate the points at $(x_j,t_n)$ by $u_j^n$. By expressing the derivatives as $$U_t\approx\frac{u_j^{n+1}-u_j^n}{k}+O(k)\text{ and }U_{xx}\approx\frac{u_{j+1}^n-2u_j^n+u_{j-1}^n}{h^2}+O(h^2),\tag2$$ we can rearrange to get $$u_j^{n+1}=(1-2r)u_j^n+r u_{j-1}^n+r u_{j+1}^n\tag3$$ where $r=k/h^2$