Please don't mark this as a duplicate question as I have read similar questions on SE but couldn't find my answer.
I have been studying numerical methods and an important part of the introduction to numerical methods is Taylor's series especially when studying numerical methods applied to D.Es(also PDEs). I have come to understand that Taylor's series expansion of a function is an expansion around a pivot point and if this pivot point is taken to be some '$a$' in the domain of the function and we have to find the value at some '$x$' close to 'a' then the expansion is given as: $$f(x)=f(a)+(x-a)f'(a)+(x-a)^2f''(a)/2!+(x-a)^3f'''(a)/3!.....$$ where $|x-a|$ can be said to be the step size.
If the pivot point is given as any '$x$' in the domain of function $f(x)$ then the approximation of the function near to the given pivot point is given by: $$f(x) or f(x+h)=f(x)+hf'(x)+h^2f''(x)/2!+h^3f'''(x)/3!.....$$ where '$h$' is the step size.
What I haven't been able to understand is how can we truncate off after a particular number of terms in the expansion...especially when the step size i.e. $|x-a|$ or $h$ is greater than $1$. I know that in the expansion of certain function we get alternating positive and negative terms...which potentially balances out...but let's consider that no negative is present in the expansion. Also, I have read that the value of the factorial increases with the order of the term and $1/factorial$ grows smaller and smaller but its growth cannot be compared to how the exponential terms grow (as here in this question the exponential is taken for the $step-size>1$).
I am studying the numerical methods to get a better grip of CFD codes. The explanations in CFD books and elsewhere the truncation-error is given in terms of the big-O notion and is always given as some exponent of the step-size value i.e if we truncate after $2nd$ order terms then the order of error is given as $O(h^3)$ (where the exponent 3 is called the order of error). Now if $h>1$ shouldn't the order of error be infinity as $h^3$ won't be the greatest term among the truncated terms any more(when h>1), but the subsequent order terms i.e. $h^4$ or $h^5$...and so on would be greater. Is this because when we use the concept of taylor's series in CFD we are assuming that the next point would be with in the radius of convergece of the polynomial after we truncate the series?