Alternating series test
In mathematical analysis, the alternating series test is the method used to show that an alternating series is convergent when its terms (1) decrease in absolute value, and (2) approach zero in the limit. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. The test is only sufficient, not necessary, so some convergent alternating series may fail the first part of the test. Formal Statement. Alternating series test. A series of the form where either all "a" are positive or all "a" are negative , is called an alternating series. The alternating series test guarantees that an alternating series converges if the following two conditions are met: Alternating series estimation theorem. 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 formula_7, where formula_8 and formula_9 for all natural numbers "n". (The case formula_10 follows by taking the negative.) Proof of the alternating series test. We will prove that both the partial sums formula_11 with odd number of terms, and formula_12 with  even number of terms, converge to the same number "L". Thus the usual partial sum formula_13 also converges to "L". The odd partial sums decrease monotonically: while the even partial sums increase monotonically: both because "a" decreases monotonically with "n". Moreover, since "a" are positive, formula_16. Thus we can collect these facts to form the following suggestive inequality: Now, note that "a" − "a" is a lower bound of the monotonically decreasing sequence "S", 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. More precisely, when there is an 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 the alternating series estimation theorem. We would like to show formula_20 by splitting into  two cases. 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. Examples. A typical example. The alternating harmonic series formula_23meets both conditions for the alternating series test and converges. An example to show monotonicity is needed. All of the conditions in the test, namely convergence to zero and monotonicity, should be met in order for the conclusion to be true . For example, take the series The signs are alternating and the terms tend to zero. However, monotonicity is not present and we cannot apply the test. Actually the series is divergent. Indeed, for the partial sum formula_25 we have formula_26 which is twice the partial sum of the harmonic series, which is divergent. Hence the original series is divergent. The test is only sufficient, not necessary. Leibniz test's monotonicity is not a necessary condition, thus the test itself is only sufficient, but not necessary. (The second part of the test is well known necessary condition of convergence for  all series.) Examples of nonmonotonic series that converge are formula_27 and formula_28