Math 221

Homework #3

Instructions. This is due on October 4 at the beginning of class. It is worth 20 points. Late papers lose 4 points if turned in Monday, plus 3 points per additional day or partial day late. You must work by yourself, but feel free to consult with me. Explain what you did and why you did it: good explanations are essential! Your paper should start with an introduction and finish with a conclusion. Start early, ask questions and do well.

Introduction. This is an exploration of one aspect of infinite series. It is frustrating (and perhaps surprising) that there are so many different tests that need to be learned. This assignment is intended to show you how difficult it is to pin down anything involving infinite series.

Problems. In terms of p-series, the following statement is true: the harmonic series is the smallest series that diverges. Briefly explain what it means for one divergent series to be “smaller” than another divergent series (after all, both series sum to infinity). For p-series, is there a largest series that converges?

Show that diverges. Compare to . Is it true in general that the harmonic series is the smallest series that diverges? Find all values of p such that diverges.

Next, show that diverges. Compare to . Find all values of p such that diverges.

The Limit Comparison Test would be the only test we needed if there were “smallest divergent” and “largest convergent” series. Explain why your work here indicates that this doesn’t happen. In particular, at any stage did you find a largest convergent series? Without showing any work, write down a divergent series that would be smaller than any of the series of the form .