Sometimes it turns out that *divergent* sums have a meaning in physics.
For example, if one adds a correction to a potential in the Schrödinger
equation, then it may be possible to compute a few terms in a perturbative
expansion (i.e. an expansion where the correction to the potential
is treated as a ``small'' perturbation)
near the original potential. It may also be possible to estimate
the behavior of the *n*'th order in the perturbative series, and the resulting
series then often turns out to be divergent. However, there does exist a
perfectly reasonable exact answer to this problem, provided the additional
potential is sufficiently well behaved.

In some cases the exact answer can be obtained from the divergent series
by a Borel summation of this series. The Borel sum can be defined the
following way: Consider the series

(63) |

which we assume to be divergent. Making use of the integral representation of

(64) |

This step is perfectly valid. The Borel summation consists in interchanging the sum and the integral. This procedure is not valid for the divergent sum from a rigorous point of view, but leads to the following definition of the Borel sum of ,

(65) |

If the series converges, the Borel sum is thus well defined. If this is not the case, the method can be repeated.

As an example, let us consider the sum

(66) |

which is obviously divergent for all . In the case of the Schrödinger equation one could e.g. imagine that

Proceeding as before, the
Borel sum is given by

(67) |

which is finite for