Alternating series—an infinite series whose terms alternate sign, i.e. whose terms are alternately positive and negative. Examples are the series 1 - 1 + 1 - 1 + 1 ... and the series 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... , which is called the alternating harmonic series.

In general, it is more difficult to determine whether a series with terms of varying sign converges or diverges than it is with a series whose terms are all of the same sign. For a series to converge, its sequence of partial sums (that is, the sequence obtained by taking just the first term, then taking the sum of the first two terms, then the sum of the first three terms, and so on) must have a limit. If all the terms in a series are positive, the partial sums will all be positive and will get larger and larger as more terms of the series are added...