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The series = + = + + is known as the alternating harmonic series. It is conditionally convergent by the alternating series test , but not absolutely convergent . Its sum is the natural logarithm of 2 .
Like any series, an alternating series is a convergent series if and only if the sequence of partial sums of the series converges to a limit. The alternating series test guarantees that an alternating series is convergent if the terms a n converge to 0 monotonically, but this condition is not necessary for convergence.
The alternating harmonic series is a classic example of a conditionally convergent series: = + is convergent, whereas = | + | = = is the ordinary harmonic series, which diverges. Although in standard presentation the alternating harmonic series converges to ln(2) , its terms can be arranged to converge to any number, or even to diverge.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A famous example of an application of this test is the alternating harmonic series = + = + +, which is convergent per the alternating series test (and its sum is equal to ), though the series formed by taking the absolute value of each term is the ordinary harmonic series, which is divergent.
If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence.
A classic example is the alternating harmonic series given by + + = = +, which converges to (), but is not absolutely convergent (see Harmonic series). Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any value at all, including ∞ or −∞; see Riemann series theorem .
A summation method that is linear and stable cannot sum the series 1 + 2 + 3 + ⋯ to any finite value. (Stable means that adding a term at the beginning of the series increases the sum by the value of the added term.) This can be seen as follows. If + + + =, then adding 0 to both sides gives