506 Chapter 8 • Infinite Series of Real NumbersCorollary 8.6.4 The set of points x for which a given power series
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L ak(x - c)k converges is a nonempty interval centered at c. That is, it must
k=O
be one of the following: { c}, JR, or a bounded interval with endpoints c - p and
c + p, for some p > 0. The series converges absolutely in the interior of this
interval, but may or may not converge at the endpoints.
Proof. Let A= { x: ~ak(x - c)k converges}' and suppose A I= {c} and
A I= JR. Then 3x 0 I= c 3 the given series diverges when x = x 0. By Corollary
8.6.3, the series diverges for all x outside the interval [c - c, c + c], where
c = lxo - cl. Hence, A ~ [c -c, c + c]. Thus, A is a bounded set. Hen ce, the set
B ={ix - cl: f ak(x - c/ converges}= {Ix - cl : x EA}
k=Ois also a bounded nonempty set. Thus, it has a least upper bound, say p =
supB. We sh all prove that (c - p, c + p) ~A~ [c - p, c + p].
(a) Suppose x E (c - p,c + p). Then Ix - cl< p = supB so 3 lx1 - cl E
B 3 Ix - cl < lx1 - cl-That is, :3x1 E A 3 Ix - cl < lx1 - cl-By Theorem 8.6.2,
t he given series converges (absolutely) at x. Therefore, (c - p, c + p) ~A.
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(b) Suppose x EA. Then Lak(x - c)k converges, so Ix - cl EB. Thus,
k=O
Ix - cl ::; sup B = p. That is ,c-p::;x::;c+p
i.e., x E [c - p , c + p].Therefore , A~ [c - p, c + p].
(c) Putting together (a) and (b) we have(c - p, c + p) ~A~ [c - p, c + p].
As noted in (a) above, the series converges absolutely for all x in ( c - p, c + p).
Definition 8.6.5 We call the set of real numbers x for which a given power
00
series L ak(x - c)k converges the interval of convergence of the series. If
k=O
that interval is bounded, we call the number p of Theorem 8.6.4 the radius of
convergence of t he series. If the series converges only at x = c we say that