An alternating series is convergent if the absolute value of any term is less than that of the preceding one and the limit of the terms is 0. For example:

S_n = -1 + 1/2 - 1/3 + 1/4 - ... + (-1)ⁿ/n

An alternating series can be treated as the sum of two series, one with only positive terms and one with only negative terms. If both are convergent separately, the alternating series is convergent. The absolute value of each term in the following alternating series is not always less than that of the preceding one. However, it is still convergent because the two series, one with only positive terms and one with only negative terms, are convergent.

S =1/2¹ -1/3⁰ + 1/2² - 1/3¹ + 1/2³ - 1/3² + 1/2⁴ - 1/3³ + ...

is convergent because both

S₁ = 1/2¹ + 1/2² + 1/2³ + 1/2⁴ + ...
and
S₂ = -1/3⁰ - 1/3¹ - 1/3² - 1/3³ - ...

are convergent.