I have a slightly daft question concerning Taylor series.
Say if I want to approximate a function about a point $a$, and I am using a finite Taylor series of order $m$, then this series will be a "decent" approximation of my function only in a small neighborhood around $a$.
However, when we are approximating functions by infinite Taylor series (which have a certain ratio of convergence $R$), we still have to define a center $a$. What I am confused about, is whether in this case the function will be "decently" approximated once again in a very small neighborhood around the center $a$ or in the whole interval, $|x-a|< R$? Is this what distinguishes the two series? Is the term approximation applicable here (for infinite series)? (Or is the function exactly equal in this interval).