Say, I have a continuos function that is infinitely differentiate on the interval $I$.
It can then be written as a taylor series. However, taylor series aren't always completely equal to the function - in other words, they don't necessarily converge for all $x$ in $I$.
Why? The way I think of taylor series is that if you know the position , velocity, acceleration, jolt etc. of a particle at one moment in time, you can calculate its position at any time. Taylor series not converging for all $x$ suggests there's a limitation on this analogy.
So why do taylor series "not" work for some $x$?
Using the particle analogy, described above shouldn't taylor series allow you to find the "location" of the function at any "time"?
Please note, I am not looking for a proof - I'm looking for an intuitive explanation of why taylor series don't always converge for all $x$.