# Having trouble understanding power series

I'm studying power series / taylor series, but I'm having understanding how the different pieces relate. I have a few questions:

1) Both the power and Taylor series has a center, but what does c represent? (i.e. if at infinity, the polynomial approximation is very close to the original function, why not just set c=0, or remove the variable altogether)

2) How does c relate to interval of convergence if at all?

3) Why does adding n-th derivative makes the polynomial a more accurate representation of the original function? (i.e. why not subtract, or multiply?)

• Important fact: the correct context for the power series is the complex analysis. – Martín-Blas Pérez Pinilla Mar 22 '17 at 15:39

1. $c$ represents the point where you know the values $f(c)$ and the derivatives $f'(c), f''(c)$ etc. The more derivatives you know, the more terms you can add to your Taylor series.
2. $c$ is the center of the interval of convergence.
• the more terms you can add to your Taylor series. Is this because there are (n-1), where n is the exponent of x, inflection points in a graph, so more terms equal greater detail? c is the center of the interval of convergence. If ratio test for convergence is -1 < c < 1, does this mean the polynomial approximation can only be applied to that interval? – Daniel Reed Mar 22 '17 at 16:16
For (1) and (2): consider the power series of $f(z) = 1/z$ around $c\ne 0$. You can check easily that the radius of convergence is $|c| =$ distance from the center to the singularity.