ABSTRACT
This chapter considers some of the subtleties of infinite series. For such series, because the partial sums form a non-decreasing sequence, convergence of the series is equivalent to the boundedness of the sequence of partial sums. Furthermore, whether or not the partial sums are bounded depends only on whether the terms of the series tend to zero quickly enough. The chapter deals with arbitrary series of real (or complex) numbers. For such series, convergence depends not just on the absolute size of the terms but also on their order; if the positive and negative terms are arranged just so, even though they do not tend to zero very quickly, there may be additional cancellation so that the partial sums nonetheless have a limit. The chapter also includes exercise problems related to infinite series and arbitrary series.
