Taylor series for different points... how do they look?

Solution 1:

...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.

Solution 2:

If you do a Taylor series around $0$ (also called a MacLaurin series) it looks like $f(x)=b_0+b_1x+b_2x^2+\ldots$. If you do it around $a$ it looks like $f(x)=b_0+b_1(x-a)+b_2(x-a)^2+\ldots$. The expansion is generally more accurate the closer $x$ is to the expansion point.