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. It predicts, that given certain conditions, all countries should converge to the same level of per capital output.

The conditions for convergence are: same population growth rates, same capital depreciation rate, same levels of education, same investment rates in physical and organizational capital, and the same rates of technological adoption. However, the Solow model fails to explain what accounts for differences in these inputs. Explanations for these differences is theorized to be explained by such causes such as climate, geography, natural endowments, economic institutions, and cultural (sociological) differences.

The Solow model also predicts that countries possessing the same rates of technological adoption 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.

See also

• Absolute convergence
• Unconditional convergence

References

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