In order to show a smooth motion, these animated gifs are fairly large. They demonstrate the development of each power series as new terms are gradually added.
A power series expansion about the point x = 0 matches the behavior of the function exactly at that point. If the series converges to the function over some finite or infinite domain, it will eventually converge to the function in that region, but how fast?
These examples show a couple of trigonometric functions, a Bessel function, and a geometric series. These power series converge for all values of the argument. The fourth example converges only up to x < 1.
Fourier Series Animations
|J0( x )|
|1/( 1 + x2 )|