Conditional convergence

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

Definition

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 finite number (not ∞ or −∞), but $\scriptstyle\sum\limits_{n=0}^\infty \left|a_n\right| = \infty.$

A classic example is the alternating series 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).

Bernhard Riemann proved that a conditionally convergent series may be rearranged to converge to any sum at all, including ∞ or −∞; see Riemann series theorem.

A typical conditionally convergent integral is that on the non-negative real axis of $\sin (x^2)$ (see Fresnel integral).

See also

• Absolute convergence
• Unconditional convergence

References

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