Alternating series test

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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.^[1][2][3]

For a generalization, see Dirichlet's test.^[4][5][6]

A series of the form

$${\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=a_{0}-a_{1}+a_{2}-a_{3}+\cdots }$$

where either all a_n are positive or all a_n 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]

1. ${\displaystyle |a_{n}|}$ decreases monotonically^[a], i.e., ${\displaystyle |a_{n+1}|\leq |a_{n}|}$, and
2. ${\displaystyle \lim _{n\to \infty }a_{n}=0}$ .

Moreover, let L denote the sum of the series, then the partial sum ${\textstyle S_{k}=\sum _{n=0}^{k}(-1)^{n}a_{n}\!}$ approximates L with error bounded by the next omitted term:

$${\displaystyle \left|S_{k}-L\right\vert \leq \left|S_{k}-S_{k+1}\right\vert =a_{k+1}.\!}$$

Suppose we are given a series of the form ${\textstyle \sum _{n=1}^{\infty }(-1)^{n-1}a_{n}\!}$, where ${\displaystyle \lim _{n\rightarrow \infty }a_{n}=0}$ and ${\displaystyle a_{n}\geq a_{n+1}}$ for all natural numbers n. (The case ${\textstyle \sum _{n=1}^{\infty }(-1)^{n}a_{n}\!}$ follows by taking the negative.)^[8]

We will prove that both the partial sums ${\textstyle S_{2m+1}=\sum _{n=1}^{2m+1}(-1)^{n-1}a_{n}}$ with odd number of terms, and ${\textstyle S_{2m}=\sum _{n=1}^{2m}(-1)^{n-1}a_{n}}$ with even number of terms, converge to the same number L. Thus the usual partial sum ${\textstyle S_{k}=\sum _{n=1}^{k}(-1)^{n-1}a_{n}}$ also converges to L.

The odd partial sums decrease monotonically:

$${\displaystyle S_{2(m+1)+1}=S_{2m+1}-a_{2m+2}+a_{2m+3}\leq S_{2m+1}}$$

while the even partial sums increase monotonically:

$${\displaystyle S_{2(m+1)}=S_{2m}+a_{2m+1}-a_{2m+2}\geq S_{2m}}$$

both because a_n decreases monotonically with n.

Moreover, since a_n are positive, ${\displaystyle S_{2m+1}-S_{2m}=a_{2m+1}\geq 0}$. Thus we can collect these facts to form the following suggestive inequality:

$${\displaystyle a_{1}-a_{2}=S_{2}\leq S_{2m}\leq S_{2m+1}\leq S_{1}=a_{1}.}$$

Now, note that a₁ − a₂ is a lower bound of the monotonically decreasing sequence S_2m+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 ${\displaystyle \lim _{m\to \infty }(S_{2m+1}-S_{2m})=\lim _{m\to \infty }a_{2m+1}=0}$.

Call the limit L, then the monotone convergence theorem also tells us extra information that

$${\displaystyle S_{2m}\leq L\leq S_{2m+1}}$$

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.

We would like to show ${\displaystyle \left|S_{k}-L\right|\leq a_{k+1}\!}$ by splitting into two cases.

When k = 2m+1, i.e. odd, then

$${\displaystyle \left|S_{2m+1}-L\right|=S_{2m+1}-L\leq S_{2m+1}-S_{2m+2}=a_{(2m+1)+1}.}$$

When k = 2m, i.e. even, then

$${\displaystyle \left|S_{2m}-L\right|=L-S_{2m}\leq S_{2m+1}-S_{2m}=a_{2m+1}}$$

as desired.

Both cases rely essentially on the last inequality derived in the previous proof.

The alternating harmonic series

$${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots }$$

meets both conditions for the alternating series test and converges.

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

$${\displaystyle {\frac {1}{{\sqrt {2}}-1}}-{\frac {1}{{\sqrt {2}}+1}}+{\frac {1}{{\sqrt {3}}-1}}-{\frac {1}{{\sqrt {3}}+1}}+\cdots \ .}$$

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 ${\textstyle S_{2n}}$ we have ${\textstyle S_{2n}={\frac {2}{1}}+{\frac {2}{2}}+{\frac {2}{3}}+\cdots +{\frac {2}{n-1}}}$ which is twice the partial sum of the harmonic series, which is divergent. Hence the original series is divergent.

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:

$${\displaystyle \sum _{n=2}^{\infty }{\frac {(-1)^{n}}{n+(-1)^{n}}}\quad {\text{and}}\quad \sum _{n=1}^{\infty }(-1)^{n}{\frac {\cos ^{2}n}{n^{2}}}\ .}$$

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.^[9]

• Alternating series
• Dirichlet's test

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