In mathematical analysis, the alternating series test is the method used to prove that an alternating series with terms that decrease in absolute value is a convergent series. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion.
Contents
Formulation
A series of the form
where either all an are positive or all an are negative, is called an alternating series.
The alternating series test then says: if
Moreover, let L denote the sum of the series, then the partial sum
approximates L with error bounded by the next omitted term:
Proof
Suppose we are given a series of the form
Proof of convergence
We will prove that both the partial sums
The odd partial sums decrease monotonically:
while the even partial sums increase monotonically:
both because an decrease monotonically with n.
Moreover, since an are positive,
Now, note that a1 − a2 is a lower bound of the monotonically decreasing sequence S2m+1, the monotone convergence theorem then implies that this sequence converges as m approaches infinity. Similarly, the sequence of even partial sum converges too.
Finally, they must converge to the same number because
Call the limit L, then the monotone convergence theorem also tells us an extra information that
for any m. This means the partial sums of an alternating series also "alternates" above and below the final limit. More precisely, when there are odd (even) number of terms, i.e. the last term is a plus (minus) term, then the partial sum is above (below) the final limit.
This understanding leads immediately to an error bound of partial sums, shown below.
Proof of partial sum error bound
We would like to show
When k = 2m+1, i.e. odd, then
When k = 2m, i.e. even, then
as desired.
Both cases rely essentially on the last inequality derived in the previous proof.
For an alternative proof using Cauchy's convergence test, see Alternating series.
For a generalization, see Dirichlet's test.