Search results
Results From The WOW.Com Content Network
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 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. An example of a conditionally convergent series is the alternating harmonic series.
The Maclaurin series of the logarithm function (+) is conditionally convergent for x = 1. The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges.
The implication of this is that poverty will ultimately disappear 'by itself'. It does not explain why some nations have had zero growth for many decades (e.g. in Sub-Saharan Africa) Conditional convergence: A country's income per worker converges to a country-specific long-run level as determined by the structural characteristics of that country.
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.
A series of real or complex numbers is said to be conditionally convergent (or semi-convergent) if it is convergent but not absolutely convergent. Conditional convergence is tested for differently than absolute convergence. One important example of a test for conditional convergence is the alternating series test or Leibniz test: [62] [63] [64 ...
What does conditionally approved mean? Applying for a loan to buy a house means meeting the standards and criteria set by a mortgage lender. That requires the fairly involved process of providing ...
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