In mathematical analysis, the alternating series test proves that an alternating series is convergent when its terms decrease monotonically in absolute value and approach zero in the limit.
The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test.
[4][5][6] Leibniz discussed the criterion in his unpublished De quadratura arithmetica of 1676[7][8] and shared his result with Jakob Hermann in June 1705[9] and with Johann Bernoulli in October, 1713.
where either all an are positive or all an are negative, is called an alternating series.
The alternating series test guarantees that an alternating series converges if the following two conditions are met:[1][2][3] Moreover, let L denote the sum of the series, then the partial sum
approximates L with error bounded by the next omitted term:
Suppose we are given a series of the form
for all natural numbers n. (The case
)[14] We will prove that both the partial sums
with odd number of terms, and
also converges to L. The odd partial sums decrease monotonically:
while the even partial sums increase monotonically:
both because an decreases monotonically with n. Moreover, since an are positive,
Thus we can collect these facts to form the following suggestive inequality:
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 extra information that
for any m. This means the partial sums of an alternating series also "alternates" above and below the final limit.
This understanding leads immediately to an error bound of partial sums, shown below.
Both cases rely essentially on the last inequality derived in the previous proof.
Philip Calabrese (1962)[15] and Richard Johnsonbaugh (1979)[16] have found tighter bounds.
meets both conditions for the alternating series test and converges.
Both conditions in the test must be met for the conclusion to be true.
The signs are alternating and the terms tend to zero.
However, monotonicity is not present and we cannot apply the test.
which is twice the partial sum of the harmonic series, which is divergent.
Hence the original series is divergent.
Examples of nonmonotonic series that converge are:
In fact, for every monotonic series it is possible to obtain an infinite number of nonmonotonic series that converge to the same sum by permuting its terms with permutations satisfying the condition in Agnew's theorem.