Conditional convergence
From Wikipedia, the free encyclopedia
In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.
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[edit] Definition
More precisely, a series 
is said to converge conditionally if 
exists and is a finite number (not ∞ or −∞), but 
A classical example is given by
which converges to 
, but is not absolutely convergent (see Harmonic series).
The simplest examples of conditionally convergent series (including the one above) are the alternating series.
Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any sum at all, including ∞ or −∞; see Riemann series theorem.
[edit] In Economics
In theory of economic growth conditional convergence means negative relationship between growth and initial level of income when countries converge to different steady states.In other words the lower the level of initial income, relative to the level of steady state the faster the economy growths. This idea is based on the assumption of diminishing returns of capital which says that countries with low level of capital per worker, in relations to their long term levels, are expected to have higher returns and hence higher rates of growth. Countries are believed to have different steady states because they are determined by factors such as population or saving rates that are specific for different countries. Neo-classical growth models introduced by Solow, Swan or Ramsey focus on the concept of conditional convergence.
[edit] See also
[edit] References
- Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).
- Robert J. Barro and Xavier Sala-i-Martin, (2004) Economic Growth, 2nd edition, Massachusetts: The MIT Press
