Operations with truncation error

Question

Assume I'm given an equation $a\frac{\partial p}{\partial x}+b\frac{\partial q}{\partial x}$. I approximate $\frac{\partial p}{\partial x}$ using a truncated taylor series to first order, such that the error is $\mathcal{O}(\Delta x^2)$. I then approximate $\frac{\partial q}{\partial x}$ using a truncated taylor series to second order, such that the error is $\mathcal{O}(\Delta x^3)$.

How does the truncation error behave when adding the two terms? Does the $\mathcal{O}(\Delta x^2)$ error from the first term impact the $\mathcal{O}(\Delta x^2)$ from the second term? Do the coefficients still get multiplied by the truncation error in these operations? What about if the orginal equation involved multiplication? Basically, I'm trying to figure out how to work with the truncation error when I have polynomials of different degrees. The error plays an important role in my analysis, so I have to be sure I carry it trough with the operations. Any good references?

Answer 1

Assuming $\Delta x\rightarrow 0$, $\mathcal{O}(\Delta x^2)+\mathcal{O}(\Delta x^3)=\mathcal{O}(\Delta x^2)$.

Of course, this is just short form for the following: let $f$ be in $\mathcal{O}(\Delta x^2)$ and $g$ be in $\mathcal{O}(\Delta x^3)$. Then, $f+g=h$ where $h$ is an element of $\mathcal{O}(\Delta x^2)$. However, this is cumbersome to write, so we use the above instead.

You should really read about Big O.

Answer 2

You already have received a good answer from @par.

Let me make this more simple : you measure the length $L_A$ of stick $A$ with an accuracy of $1$ inch and the length $L_B$ of stick $B$ with an accuracy of $0.1$ inch.

Answer the question : what is the accuracy for $L_A+L_B$ ? $1$ inch, isn't it ?

This is very similar to your problem $O\left(x^1\right)+O\left(x^2\right)=O\left(x^1\right)$