A power series is always of the form . That sum is "about x= c". Once can also show that a power series always has a "radius of convergence", a number, r, such that the series converges for x in (c- r, c+ r) and diverges outside [c- r, c+ r] (may or may not converge at the endpoints, c- r and c+ r) (for real coefficients- if we allow complex numbers, the "interval of convergence" is the disk and so really is a "radius").
If you are actually talking about "series"- the infinite sums, then, no, being further from c does not affect "accuracy". For a function "analytic" inside it radius of convergence of the entire infinite sum gives the exact function. (There are functions, f(x), that have Taylor's series that converges for all x (the "radius of convergence is infinite) but NOT to f(x). The simplest example is " if x is not 0, f(0)= 0".)
Of course, if you are using finite sums, to nth power, to approximate a function, then the Lagrange error formula is so that, yes, if you are further from c, the error may be greater because is larger. But that is an upper bound on the error- not necessarily the error.