Alternating series test

Leibniz discussed the criterion in his unpublished De quadratura arithmetica of 1676{{Cite journal |last=Beeley |first=Philip |date=1995-07-01 |title=Leibniz: De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis |url=https://www.pdcnet.org/leibniz/content/leibniz_1995_0005_0000_0015_0017?file_type=pdf |journal=The Leibniz Review |language=en |volume=5 |pages=15–17 |doi=10.5840/leibniz1995517|url-access=subscription }}{{Cite book |last=Liebniz |title=De quadratura arithmetica |at=Proposition 49}} and shared his result with Jakob Hermann in June 1705{{Cite book |last=Knopp |first=Konrad |url=https://books.google.com/books?id=HPkgAAAAMAAJ&q=alternating |title=Theory and Application of Infinite Series |date=1928 |publisher=Blackie & Son |pages=131 |language=en}} and with Johann Bernoulli in October, 1713.{{Cite book |last=Ferraro |first=Giovanni |url=https://books.google.com/books?id=vLBJSmA9zgAC&q=leibniz%20bernoulli |title=The Rise and Development of the Theory of Series up to the Early 1820s |date=2007-12-20 |publisher=Springer Science & Business Media |isbn=978-0-387-73468-2 |series=Sources and Studies in the History of Mathematics and Physical Sciences |language=en}} It was only formally published in 1993.{{Cite journal |last=Knobloch |first=Eberhard |date=2006-02-01 |title=Beyond Cartesian limits: Leibniz's passage from algebraic to "transcendental" mathematics |url=https://www.sciencedirect.com/science/article/pii/S0315086004000199?ref=pdf_download&fr=RR-2&rr=90cd2cc71809e5e6 |journal=Historia Mathematica |series=The Origins of Algebra: From al-Khwarizmi to Descartes |volume=33 |issue=1 |pages=113–131 |doi=10.1016/j.hm.2004.02.001 |issn=0315-0860|url-access=subscription }}{{Cite book |last=Leibniz |first=Gottfried Wilhelm |title=De quadratura arithmetica circuli ellipseos et hyperbolae cujus corollarium est trigonometria sine tabulis |publisher=Vandenhoeck & Ruprecht |year=1993 |isbn=978-3-525-82120-6 |editor-last=Knobloch |editor-first=Eberhard |editor-link=Eberhard Knobloch |series=Abhandlungen der Akademie der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse |volume=043 |location=Göttingen |language=la, de |trans-title=On the arithmetical quadrature of the circle, the ellipse, and the hyperbola, whose corollary is trigonometry without tables |orig-year=1676 }}

A series of the form

$$\backslash sum\_\{n=0\}^\backslash infty\; (-1)^\{n\}\; a\_n\; =\; a\_0-a\_1\; +\; a\_2\; -\; a\_3\; +\; \backslash cdots$$

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. $|a\_n|$ decreases monotonically,{{efn|In practice, the first few terms may increase. What is important is that $b\_\{n\}\; \backslash geq\; b\_\{n+1\}$ for all $n$ after some point,{{cite web |last1=Dawkins |first1=Paul |title=Calculus II - Alternating Series Test |url=http://tutorial.math.lamar.edu/Classes/CalcII/AlternatingSeries.aspx |website=Paul's Online Math Notes |publisher=Lamar University |access-date=1 November 2019}} because the first finite amount of terms would not change a series' convergence/divergence.}} i.e., $|a\_\{n+1\}|\backslash leq|a\_n|$, and
2. $\backslash lim\_\{n\; \backslash to\; \backslash infty\}\; a\_n\; =\; 0$ .

Moreover, let L denote the sum of the series, then the partial sum $S\_k\; =\; \backslash sum\_\{n=0\}^k\; (-1)^\{n\}\; a\_n\backslash !$ approximates L with error bounded by the next omitted term:

$$\backslash left\; |\; S\_k\; -\; L\; \backslash right\; \backslash vert\; \backslash le\; \backslash left\; |\; S\_k\; -\; S\_\{k+1\}\; \backslash right\; \backslash vert\; =\; a\_\{k+1\}.\backslash !$$

Suppose we are given a series of the form $\backslash sum\_\{n=1\}^\backslash infty\; (-1)^\{n-1\}\; a\_n\backslash !$, where $\backslash lim\_\{n\backslash rightarrow\backslash infty\}a\_\{n\}=0$ and $a\_n\; \backslash geq\; a\_\{n+1\}$ for all natural numbers n. (The case $\backslash sum\_\{n=1\}^\backslash infty\; (-1)^\{n\}\; a\_n\backslash !$ follows by taking the negative.)The proof follows the idea given by James Stewart (2012) “Calculus: Early Transcendentals, Seventh Edition” pp. 727–730. {{ISBN|0-538-49790-4}}

{{for|an alternative proof using Cauchy's convergence test|Alternating series#Alternating series test}}

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

The odd partial sums decrease monotonically:

$$S\_\{2(m+1)+1\}=S\_\{2m+1\}-a\_\{2m+2\}+a\_\{2m+3\}\; \backslash leq\; S\_\{2m+1\}$$

while the even partial sums increase monotonically:

$$S\_\{2(m+1)\}=S\_\{2m\}+a\_\{2m+1\}-a\_\{2m+2\}\; \backslash geq\; S\_\{2m\}$$

both because a_n decreases monotonically with n.

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

$$a\_1\; -\; a\_2\; =\; S\_2\; \backslash leq\; S\_\{2m\}\; \backslash leq\; S\_\{2m+1\}\; \backslash leq\; S\_1\; =\; a\_1\; .$$

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 $\backslash lim\_\{m\backslash to\backslash infty\}(S\_\{2m+1\}-S\_\{2m\})=\backslash lim\_\{m\backslash to\backslash infty\}a\_\{2m+1\}=0$.

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

$$S\_\{2m\}\; \backslash leq\; L\; \backslash leq\; S\_\{2m+1\}$$

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 $\backslash left|\; S\_k\; -\; L\; \backslash right|\; \backslash leq\; a\_\{k+1\}\backslash !$ by splitting into two cases.

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

$$\backslash left|\; S\_\{2m+1\}\; -\; L\; \backslash right|\; =\; S\_\{2m+1\}\; -\; L\; \backslash leq\; S\_\{2m+1\}\; -\; S\_\{2m+2\}\; =\; a\_\{(2m+1)+1\}\; .$$

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

$$\backslash left|\; S\_\{2m\}\; -\; L\; \backslash right|\; =\; L\; -\; S\_\{2m\}\; \backslash leq\; S\_\{2m+1\}\; -\; S\_\{2m\}\; =\; a\_\{2m+1\}$$

as desired.

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

Philip Calabrese (1962){{Cite journal |last=Calabrese |first=Philip |date=1962 |title=A Note on Alternating Series |url=https://www.jstor.org/stable/2311056 |journal=The American Mathematical Monthly |volume=69 |issue=3 |pages=215–217 |doi=10.2307/2311056 |jstor=2311056 |issn=0002-9890|url-access=subscription }} and Richard Johnsonbaugh (1979){{Cite journal |last=Johnsonbaugh |first=Richard |date=1979-10-01 |title=Summing an Alternating Series |url=https://www.tandfonline.com/doi/abs/10.1080/00029890.1979.11994875 |journal=The American Mathematical Monthly |volume=86 |issue=8 |pages=637–648 |doi=10.1080/00029890.1979.11994875 |issn=0002-9890|url-access=subscription }} have found tighter bounds.{{Cite journal |last=Villarino |first=Mark B. |date=2018 |title=The Error in an Alternating Series |url=https://www.jstor.org/stable/48663300 |journal=The American Mathematical Monthly |volume=125 |issue=4 |pages=360–364 |doi=10.1080/00029890.2017.1416875 |jstor=48663300 |hdl=10669/75532 |issn=0002-9890|arxiv=1511.08568 }}

The alternating harmonic series

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

meets both conditions for the alternating series test and converges.

Both conditions in the test must be met for the conclusion to be true. For example, take the series

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

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 $S\_\{2n\}$ we have $S\_\{2n\}=\backslash frac\{2\}\{1\}+\backslash frac\{2\}\{2\}+\backslash frac\{2\}\{3\}+\backslash cdots+\backslash frac\{2\}\{n-1\}$ 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.

Examples of nonmonotonic series that converge are:

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

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.{{cite journal |first=Ralph Palmer |last=Agnew |date=1955 |title=Permutations preserving convergence of series |url=https://www.ams.org/journals/proc/1955-006-04/S0002-9939-1955-0071559-4/S0002-9939-1955-0071559-4.pdf |journal=Proc. Amer. Math. Soc. |volume=6 |issue=4 |pages=563–564|doi=10.1090/S0002-9939-1955-0071559-4 }}

• Alternating series
• Dirichlet's test

{{notelist}}

{{Reflist}}

• {{Cite book

| last = Apostol | first = Tom M. | author-link = Tom M. Apostol

| year = 1967 |orig-year=1961

| title = Calculus

| volume = 1

| edition = 2nd

| publisher = John Wiley & Sons

| isbn = 0-471-00005-1

}}

• Konrad Knopp (1956) Infinite Sequences and Series, § 3.4, Dover Publications {{ISBN|0-486-60153-6}}
• Konrad Knopp (1990) Theory and Application of Infinite Series, § 15, Dover Publications {{ISBN|0-486-66165-2}}
• {{Cite book |last=Rudin |first=Walter |author-link=Walter Rudin |title=Principles of mathematical analysis |publisher=McGraw-Hill |year=1976 |isbn=0-07-054235-X |edition=3rd |location=New York |oclc=1502474 |orig-date=1953}}
• {{Cite book

|last=Spivak |first=Michael | author-link = Michael Spivak

|year=2008 |orig-year=1967

|title=Calculus

|edition=4th

|location=Houston, TX |publisher=Publish or Perish

|isbn=978-0-914098-91-1

}}

• James Stewart, Daniel Clegg, Saleem Watson (2016) Single Variable Calculus: Early Transcendentals (Instructor's Edition) 9E, Cengage ISBN 978-0-357-02228-9
• E. T. Whittaker & G. N. Watson (1963) A Course in Modern Analysis, 4th edition, §2.3, Cambridge University Press {{ISBN|0-521-58807-3}}

{{Gottfried Wilhelm Leibniz}}

{{DEFAULTSORT:Alternating Series Test}}

Category:Convergence tests

Category:Gottfried Wilhelm Leibniz