A more powerful alternating series theorem

In application you often truncate an infinite series to give a practical approximation. Ideally you’d like to know how accurate the approximation is. It would be even better to know the sign of the error of the approximation.

Alternating series let you do this. But some forms of the alternating series theorem leave money on the table. That is, a more general and more useful form of the theorem is possible.

Weak statement

Here’s a weak statement of the alternating series theorem taken from a textbook.

If a₁ ≥ a₂ ≥ a₃ ≥ … ≥ a_n ≥ … 0 for all n, then the alternating series

a₁ − a₂ + a₃ − a₄ + …

converges to a real number S. Moreover, if the nth partial sums is denoted S_n then

|S − S_n| ≤ S_n+1.

The notation above is a little vague when it says “… 0”. The theorem requires that all the a‘s are non-negative and that their limit is 0.

Small improvements

One easy improvement is to note that it’s only necessary for the terms to eventually satisfy the conditions of the theorem. If the terms don’t initially decrease in absolute value, or don’t initially alternate in sign, then apply the theorem starting after a point where the terms do decrease in absolute value and alternate in sign.

Another improvement is to note that we can tell the sign of the truncation error. That is, if S_n+1 is positive, then

0 ≤ S − S_n ≤ S_n+1,

and if S_n+1 is negative, then

S_n+1 ≤ S − S_n ≤ 0.

Stronger statement

A stronger statement of the alternating series theorem is to note that it is not necessary for all the terms to alternate. Also, you can have truncation error estimates even if the series doesn’t converge.

Here is a much stronger statement of the alternating series theorem, taken from the book Interpolation by J. F. Steffensen, with notation changed to match the theorem above.

Let

S = a₁ + a₂ + … + a_n + R_n+1

and let the first non-zero term after a_n be a_n+s. If R_n+1 and R_n+s+1 have opposite signs then the remainder term is numerically smaller than the first rejected non-zero term, and has the same sign.

Note that there’s no issue of convergence, and so the theorem could be applied to a divergent series. And there’s no requirement that the terms alternate signs, only that the specified remainder terms alternate signs.

Applications

A lot of functions that come up in applications have alternating series representations. If a series does not alternate, it can be useful to look for a reformulation that does have an alternating series, because, as explained here, truncation error is easy to estimate. Alternating series can also often be accelerated.

The next post shows that the error function, a slight variation on the Gaussian cumulative distribution function, has both an alternating power series and an alternating asymptotic series.