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

A classical example is given by

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

which converges to Ln 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.

See also

• Absolute convergence
• Unconditional convergence

References

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