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In particular, for series with values in any Banach space, absolute convergence implies convergence. The converse is also true: if absolute convergence implies convergence in a normed space, then the space is a Banach space. If a series is convergent but not absolutely convergent, it is called conditionally convergent.
More precisely, a series of real numbers = is said to converge conditionally if = exists (as a finite real number, i.e. not or ), but = | | =.. A classic example is the alternating harmonic series given by + + = = +, which converges to (), but is not absolutely convergent (see Harmonic series).
"Beta-convergence" on the other hand, occurs when poor economies grow faster than rich ones. Economists say that there is "conditional beta-convergence" when economies experience "beta-convergence" but conditional on other variables (namely the investment rate and the population growth rate) being held constant.
3 Conditional and absolute convergence. 4 Uniform convergence. ... The series can be compared to an integral to establish convergence or divergence. Let () = ...
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.
In a normed vector space, one can define absolute convergence as convergence of the series (| |). Absolute convergence implies Cauchy convergence of the sequence of partial sums (by the triangle inequality), which in turn implies absolute convergence of some grouping (not reordering). The sequence of partial sums obtained by grouping is a ...
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series =. List of tests
Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence () =, with {, +}, the series = converges. If X {\displaystyle X} is a Banach space , every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general.