TITLE: Taylor series for different points... how do they look?
QUESTION [6 upvotes]: I can't understand what it means to do the Taylor series at the point $a$.
The best way would be showing me how it looks for different $a$ on a graph. Do I find those graphs on the Internet?

REPLY [20 votes]: ...The best way would be showing me how it looks for different $a$ on a graph.

The others have done (most of) the math; I'll do the cartoons:
$\exp\,x$:

$\dfrac1{1-x}$:

$\ln(1+x)$:

$\arctan\,x$:

$\sin\,x$:


Note that the polynomials (except for the horizontal constant function) are "tangent" to the original function at the expansion point (shown in red above). That's sort of the idea: these polynomials are the unique $p$-th degree polynomials that have $p+1$-fold contact with the function being approximated.