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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. This is because a power series is absolutely convergent on the interior of its disk of convergence. [a]
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 .
Every absolute convergent series (real or complex) is also convergent, but the converse is not true. The Maclaurin series of the exponential function is absolutely convergent for every complex value of the variable. If the series = converges but the series = | | diverges, then the series = is conditionally convergent.
The convergence of each absolutely convergent series is an equivalent condition for a normed vector space to be Banach (i.e.: complete). Absolute convergence and convergence together imply unconditional convergence, but unconditional convergence does not imply absolute convergence in general, even if the space is Banach, although the ...
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the Riemann series theorem. [31] [32] [33]
If < then the series converges absolutely. If ℓ ≤ 1 ≤ L {\displaystyle \ell \leq 1\leq L} then the test is inconclusive, and the series may converge absolutely, conditionally or diverge. Root test
Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence () =, with {, +}, the series = converges.. If is a Banach space, every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general.