Download MATH 4335 – Shipman Study Problems 9 Prove your answers! 1

MATH 4335 – Shipman
Study Problems 9
Prove your answers!
1. Domain of convergence of a power series
Study Section 6.5 (Power Series)
a) What is a power series, and how does it define a function? Give some examples of functions
that are defined as power series – you will find many from your study of Calculus II!
b) The domain on which a power series converges has a very special form. What is it? Prove your
answer. What key result about series is used to prove this?
c) What does it mean to speak of the “radius of convergence” R of a power series? Can the radius
of convergence be infinite? Can it be zero? If so, find examples. Find an example of a power series
where R = 1 and examples where R = 2 and R = 105.
d) If a power series converges at x = c, does it also converge at x = −c? If it converges at −c, does
it also converge at c?
e) What is the difference between saying that a power series converges at p and that it converges
absolutely at p? Give an example of a power series that converges at some point p but does not
converge absolutely there. If a power series converges absolutely at x = 1, does it also converge at
x = −1? If a power series converges absolutely at x = −1, does it also converge at x = 1?
2. Uniform convergence and continuity of power series
a) If a power series converges on an interval I, must it converge uniformly on I? As always, either
prove that it does or find a counterexample.
b) If a power series converges at a point p, must it converge uniformly on the interval [−|p|, |p|]?
Does the outcome change if the power series converges absolutely at p? Prove Theorem 6.5.2!
What major theorem is helpful in proving this?
c) Theorem 6.5.5 is a slightly stronger statement than the result we proved in (b) (see Theorem
6.5.2). In what way is Theorem 6.5.5 stronger than 6.5.2, and what extra work is needed to obtain
the improvement?
d) Show how we can use Theorem 6.5.2, together with another result that we proved earlier, to
conclude that if a power series converges on an open interval (−R, R), then it is continuous on that
whole interval. What if the power series converges also at R or −R? May we conclude in the same
way that the series is continuous on its whole interval of convergence? Is there an example where
a power series converges at an endpoint of its interval of convergence but is not continuous there?
Explain.
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3. Special examples
a) Work out the following problems, involving interesting examples and counter-examples:
Problems 6.4.5, 6.5.1, 6.5.2
b) Study again Section 6.1, this time with a view toward the results about series that we have
studied so far.
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