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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.
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.
In the following, a sum or product taken over p always represents a sum or product taken over a specified set of primes. The proof rests upon the following four inequalities: Every positive integer i can be uniquely expressed as the product of a square-free integer and a square as a consequence of the fundamental theorem of arithmetic .
= + = + +, which has a sum of the natural logarithm of 2, while the sum of the absolute values of the terms is the harmonic series, = = + + + + +, which diverges per the divergence of the harmonic series, [28] so the alternating harmonic series is conditionally convergent.
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 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 .