False vacuum

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 $\sum _{n=0}^{\mathrm{\infty }}{a}_n$ is said to converge conditionally if $\underset{m\to \mathrm{\infty }}{lim}\sum _{n=0}^m{a}_n$ exists and is a finite number (not ∞ or −∞), but $\sum _{n=0}^{\mathrm{\infty }}\left|a_n\right|=\mathrm{\infty }.$

A classic example is given by

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

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.

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

See also

• Absolute convergence
• Unconditional convergence

References

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