ABSTRACT
A series is an infinite sum. People think of the series as the limit of the sequence of its partial sums. While at first a bit confusing, this approach avoids many conundrums and redundancies that have plagued the history of series. A series is an infinite sum. One of the most effective ways to handle an infinite process in mathematics is with a limit. The harmonic series diverges, but it diverges very slowly. Series with nonnegative summands are generally much easier to understand, and to analyze, than series with both positive and negative summands. This is because series of the first type converge because of size of the terms alone. But series of the second type can and do converge because of cancellation. One of the elegant features of the theory of numerical series is that there are several convergence tests that are easy to apply to get specific, concrete information about convergence.
