Conditional convergence

In mathematics, a series or integral is said to be conditionally convergent if it converges but does not converge absolutely.

More precisely, a series $\scriptstyle \sum _{n=0}^{\infty }a_{n}$ is said to converge conditionally if $\scriptstyle \lim _{m\rightarrow \infty }\,\sum _{n=0}^{m}\,a_{n}$ exists and is a real number (not ∞ or −∞), but $\scriptstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=\infty .$

A classical example is given by

1-{1 \over 2}+{1 \over 3}-{1 \over 4}+{1 \over 5}-\cdots =\sum _{n=1}^{\infty }{(-1)^{n+1} \over n}

which converges to log_e 2, 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.

Economics

Conditional convergence is predicted by the Solow model. Conditional convergence states that countries that possess the same technological possibilities, and the same ability and potential to effectively incorporate technology in the economy, will converge to the same rate of output growth per capita. Thus, economic growth in the long run will be independent of the population growth rate, the savings rate, the rate of capital depreciation, and the initial capital-output ratio.

Further, it implies that difference in the long run growth rate of output per capita can be explained fully by exogenous variables such as geography, and by cultural, political and socio-economic institutions. Thus, to manipulate a country's long run growth rate requires that one addresses the cultural, political and socio-economic institutions of the country.

References

Walter Rudin, Principles of Mathematical Analysis (McGraw-Hill: New York, 1964).

Economics

See also

References