Question:
Why is the radius of convergence of the first 6 terms of the power series expansion of x^(1/2) centered at 4 less than 6?

Hi Nowl
The term "Radius of Convergence" applies to the entire series not just the first few terms. For your power series expansion of x^1/2 centered at 4, if r is the radius of convergence then for any x which is less than r units from 4, ie. |x-4| < r, the series converges to x^1/2. Also if |x-4| > r then the series diverges. If r is larger than 4 then there is a negative number x that satisfies |x-4| < r and hence a negative number x for which the power series converges to x^1/2. But x is negative and thus x^1/2 is not a real number. Thus the radius of convergence of the power series is less than of equal to 4 and hence less than 6.

Cheers
Harley