Search results
Results from the WOW.Com Content Network
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 .
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.
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 .
The case of =, = yields the harmonic series, which diverges. The case of =, = is the ... are convergent series. This test is also known as the Leibniz ...
by the divergence of the harmonic series. This shows that x k ≥ 1 {\displaystyle x_{k}\geq 1} for all k {\displaystyle k} , and since the tails of a convergent series must themselves converge to zero, this proves divergence.
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 series is convergent (or converges) if and only if the sequence ... The reciprocals of the positive integers produce a divergent series (harmonic series):
= + = + +, 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.