Problem Statement: Consider the power series
(a) For which values of does it converge absolutely? Conditionally? (b) Show that on the convergence is uniform.
(a) Claim: the series converges absolutely on and conditionally on .
Consider , then it follows that and so we may find a rational . Then it follows that for every . By the geometric series test converges and so by the comparison test converges. Thus, the series converges absolutely on .
When then the series becomes , the alternating harmonic series, which we know converges by the AST. But when the series is simply the harmonic series which we know diverges. Thus, the series converges conditionally on .
(b) Proof: Consider the series above for it follows that
for all and for all
We know that converges by the p-series test (since ). And so applying the M-Test we may conclude that converges uniformly on .
Reflection: This was a great reminder problem. Whenever I see uniform convergence my first thought is usually to try using the definition, which can be a real pain in the butt. This problem reminded me that when you’re dealing with uniform convergence you should try to use the M-Test if you can because it is usually much much simpler than trying to go straight from the definition. Try the easy way first!!!