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12.8   Theorem. Let be a power series. Suppose converges for some . Then converges absolutely for all .

Proof: Since converges, is a null sequence, and hence is bounded. Say for all . Let , so , and let . Then for all

Now is a convergent geometric series, so by the comparison test, converges; i.e., is absolutely convergent.

12.9   Corollary. Let be a power series. Suppose diverges for some . Then diverges for all with .

Proof: Suppose . If converges, then by the theorem, would also converge, contrary to our assumption.

12.10   Theorem. Let be a power series. Then one of the following three conditions holds:
a) converges only when .
b) converges for all .
c) There is a number such that converges absolutely for and diverges for .

Proof: Suppose that neither a) nor b) is true. Then there are numbers such that converges and diverges. If , and , it follows that converges and diverges. By a familiar procedure, build a binary search sequence such that , and for all , converges and diverges. Let be the number such that . Then for all and . If , then for some we have , and

If , then for some we have , and

12.11   Definition (Radius of convergence.) Let be a power series. If there is a number such that converges for , and diverges for , we call the radius of convergence of . If converges only for , we say has radius of convergence . If converges for all , we say has radius of convergence .

If a power series has radius of convergence , I call the disc of convergence for the series, and I call the circle of convergence for the series. If , I call the disc of convergence of the series (even though is not a disc).

12.12   Example. I will find the radius of convergence for . I will apply the ratio test. Since the ratio test applies to positive sequences, I will consider absolute convergence. Let for all . Then for all ,

Hence

By the ratio test, is absolutely convergent if , and is divergent if . It follows that the radius of convergence for our series is .

12.13   Exercise. Find the radius of convergence for the following power series:
a)
.
b)
.

12.14   Exercise. A Let be a positive real number.
a)
Find a power series whose radius of convergence is equal to .
b)
Find a power series whose radius of convergence is .
c)
Find a power series whose radius of convergence is .

Next: 12.3 Differentiation of Power Up: 12. Power Series Previous: 12.1 Definition and Examples   Index