Date: Mon, 19 Apr 1999 21:38:33 0600 (CST)
Name: Nowl
Who is asking: Student
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. x4 < r, the series converges to x^{1/2}. Also if x4 > r then the series diverges. If r is larger than 4 then there is a negative number x that satisfies x4 < 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
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