View source for Convergence tests

{{Short description|Mathematical criterion about whether a series converges}}
{{Calculus |Series}}

In [[mathematics]], '''convergence tests''' are methods of testing for the [[Convergent series|convergence]], [[conditional convergence]], [[absolute convergence]], [[interval of convergence]] or divergence of an [[series (mathematics)|infinite series]] <math>\sum_{n=1}^\infty a_n</math>.

==List of tests==

===[[Term test|Limit of the summand]]===
If the limit of the summand is undefined or nonzero, that is <math>\lim_{n \to \infty}a_n \ne 0</math>, then the series must diverge. In this sense, the partial sums are [[Cauchy sequence|Cauchy]] [[only if]] this limit exists and is equal to zero. The test is inconclusive if the limit of the summand is zero. This is also known as the '''nth-term test''', '''test for divergence''', or '''the divergence test'''.

===[[Ratio test]]===
This is also known as '''d'Alembert's criterion'''.

: Consider two limits <math>\ell=\liminf_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|</math> and <math>L=\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|</math>. If <math>\ell>1</math>, the series diverges. If <math>L<1</math> then the series converges absolutely. If <math>\ell\le1\le L</math> then the test is inconclusive, and the series may converge absolutely, conditionally or diverge.

===[[Root test]]===
This is also known as the '''''n''th root test''' or '''Cauchy's criterion'''.

: Let
:: <math>r=\limsup_{n\to\infty}\sqrt[n]{|a_n|},</math>
: where <math>\limsup</math> denotes the [[limit superior]] (possibly <math>\infty</math>; if the limit exists it is the same value).
: If ''r'' < 1, then the series converges absolutely. If ''r'' > 1, then the series diverges. If ''r'' = 1, the root test is inconclusive, and the series may converge or diverge.
The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely.<ref>{{cite web|url=http://www.mathcs.org/analysis/reals/numser/t_ratio.html|title=MathCS.org - Real Analysis: Ratio Test|first=Bert G.|last=Wachsmuth|website=www.mathcs.org}}</ref>

===[[Integral test]]===

The series can be compared to an integral to establish convergence or divergence. Let <math>f:[1,\infty)\to\R_+</math> be a non-negative and [[monotonic function|monotonically decreasing function]] such that <math>f(n) = a_n</math>.  If
<math display="block">\int_1^\infty f(x) \, dx=\lim_{t\to\infty}\int_1^t f(x) \, dx<\infty,</math>
then the series converges. But if the integral diverges, then the series does so as well. 
In other words, the series <math>{a_n}</math> converges [[if and only if]] the integral converges.

===={{mvar|p}}-series test====
A commonly-used corollary of the integral test is the p-series test. Let <math>k > 0</math>. Then <math>\sum_{n=k}^{\infty} \bigg(\frac{1}{n^p}\bigg)</math> converges if <math>p > 1</math>.

The case of <math>p = 1, k = 1</math> yields the harmonic series, which diverges. The case of <math>p = 2, k = 1</math> is the [[Basel problem]] and the series converges to <math>\frac{\pi^2}{6}</math>. In general, for <math>p > 1, k = 1</math>, the series is equal to the [[Riemann zeta function]] applied to <math>p</math>, that is <math>\zeta(p)</math>.

===[[Direct comparison test]]===
If the series <math>\sum_{n=1}^\infty b_n</math> is an [[absolutely convergent]] series and <math>|a_n|\le |b_n|</math> for sufficiently large ''n''&nbsp;, then the series <math>\sum_{n=1}^\infty a_n</math> converges absolutely.

===[[Limit comparison test]]===
If <math>\{a_n\},\{b_n\}>0</math>, (that is, each element of the two sequences is positive) and the limit <math>\lim_{n\to\infty} \frac{a_n}{b_n}</math> exists, is finite and non-zero, then either both series converge or both series diverge.

===[[Cauchy condensation test]]===

Let <math>\left \{ a_n \right \}</math> be a non-negative non-increasing sequence. Then the sum <math>A = \sum_{n=1}^\infty a_n</math> converges [[if and only if]] the sum <math>A^* = \sum_{n=0}^\infty 2^n a_{2^n}</math> converges. Moreover, if they converge, then <math>A \leq A^* \leq 2A</math> holds.

===[[Abel's test]]===

Suppose the following statements are true:

# <math>\sum a_n </math> is a convergent series,
# <math>\left\{b_n\right\}</math> is a monotonic sequence, and
# <math>\left\{b_n\right\}</math> is bounded.

Then <math>\sum a_nb_n </math> is also convergent.

===[[Absolute convergence|Absolute convergence test]]===

Every [[absolutely convergent]] series converges.

===[[Alternating series test]]===

Suppose the following statements are true:

* <math> (a_n)_{n=1}^\infty </math> is monotonic,
* <math> \lim_{n \to \infty} a_n = 0 </math>
Then <math> \sum_{n = 1}^\infty (-1)^{n} a_n </math> and <math> \sum_{n = 1}^\infty (-1)^{n+1} a_n </math> are convergent series.  
This test is also known as the '''Leibniz criterion'''.

===[[Dirichlet's test]]===

If <math>\{a_n\}</math> is a [[sequence]] of [[real number]]s and <math>\{b_n\}</math> a sequence of [[complex number]]s satisfying

:* <math>a_n \geq a_{n+1}</math>

:* <math>\lim_{n \rightarrow \infty}a_n = 0</math>

:* <math>\left|\sum^{N}_{n=1}b_n\right|\leq M</math> for every positive integer ''N''

where ''M'' is some constant, then the series

:<math>\sum^{\infty}_{n=1}a_n b_n</math>

converges.

===[[Cauchy's convergence test]]===

A series <math>\sum_{i=0}^\infty a_i</math> is convergent if and only if for every <math>\varepsilon>0</math> there is a natural number ''N'' such that

:<math>|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon</math>

holds for all {{nobreak|''n'' > ''N''}} and all {{nobreak|''p'' ≥ 1}}.

===[[Stolz–Cesàro theorem]]===

Let  <math>(a_n)_{n \geq 1}</math> and  <math>(b_n)_{n \geq 1}</math> be two sequences of real numbers. Assume that <math>(b_n)_{n \geq 1}</math> is a [[Monotonic function|strictly monotone]] and divergent sequence and the following limit exists:
:<math> \lim_{n \to \infty} \frac{a_{n+1}-a_n}{b_{n+1}-b_n}=l.\ </math>
Then, the limit
:<math> \lim_{n \to \infty} \frac{a_n}{b_n}=l.\ </math>

===[[Weierstrass M-test]]===

Suppose that (''f''<sub>''n''</sub>) is a sequence of real- or complex-valued functions defined on a set ''A'', and that there is a sequence of non-negative numbers (''M''<sub>''n''</sub>) satisfying the conditions
* <math>|f_n(x)|\leq M_n</math> for all <math>n \geq 1</math> and all <math>x \in A</math>, and
* <math>\sum_{n=1}^{\infty} M_n </math> converges.
Then the series 
:<math>\sum_{n=1}^{\infty} f_n (x)</math>
converges absolutely and [[uniform convergence|uniformly]] on ''A''.

===Extensions to the ratio test===

The ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allows one to deal with this case.

====[[Ratio test#2. Raabe's test|Raabe–Duhamel's test]]====

Let { ''a''<sub>''n''</sub> } be a sequence of positive numbers.

Define

:<math>b_n=n\left(\frac{a_n}{a_{n+1}}-1 \right).</math>

If

:<math>L=\lim_{n\to\infty}b_n</math>

exists there are three possibilities:

* if ''L'' > 1 the series converges (this includes the case ''L'' = ∞)
* if ''L'' < 1 the series diverges
* and if ''L'' = 1 the test is inconclusive.

An alternative formulation of this test is as follows. Let {{nowrap|{ ''a''<sub>''n''</sub> }}} be a series of real numbers. Then if ''b'' > 1 and ''K'' (a natural number) exist such that

:<math>\left|\frac{a_{n+1}}{a_n}\right|\le 1-\frac{b}{n} </math>

for all ''n'' > ''K'' then the series {''a''<sub>''n''</sub>} is convergent.

====[[Ratio test#3. Bertrand's test|Bertrand's test]]====

Let { ''a''<sub>''n''</sub> } be a sequence of positive numbers.

Define

:<math>b_n=\ln n\left(n\left(\frac{a_n}{a_{n+1}}-1 \right)-1\right).</math>

If

:<math>L=\lim_{n\to\infty}b_n</math>

exists, there are three possibilities:<ref>František Ďuriš, ''[http://oldwww.dcs.fmph.uniba.sk/bakalarky/obhajene/getfile.php/new.pdf?id=90&fid=228&type=application%2Fpdf Infinite series: Convergence tests]'', pp. 24–9.  Bachelor's thesis.</ref><ref>{{Cite web|url=https://mathworld.wolfram.com/BertrandsTest.html|title=Bertrand's Test|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2020-04-16}}</ref>

* if ''L'' > 1 the series converges (this includes the case ''L'' = ∞)
* if ''L'' < 1 the series diverges
* and if ''L'' = 1 the test is inconclusive.

====[[Ratio test#5. Gauss's test|Gauss's test]]====
Let { ''a''<sub>''n''</sub> } be a sequence of positive numbers.  If <math>\frac{a_n}{a_{n + 1}} = 1+ \frac{\alpha}{n} + O(1/n^\beta)</math> for some β > 1, then <math> \sum a_n</math> converges if {{math|α > 1}} and diverges if {{math|α ≤ 1}}.<ref>* {{SpringerEOM |id=Gauss_criterion&oldid=39905 |title=Gauss criterion}}</ref>

====[[Ratio test#6. Kummer's test|Kummer's test]]====
Let { ''a''<sub>''n''</sub> } be a sequence of positive numbers. Then:<ref>{{Cite journal|date=1835-01-01|title=Über die Convergenz und Divergenz der unendlichen Reihen.|url=https://www.degruyter.com/document/doi/10.1515/crll.1835.13.171/html|journal=Journal für die reine und angewandte Mathematik|volume=1835|issue=13|pages=171–184|doi=10.1515/crll.1835.13.171|s2cid=121050774|issn=0075-4102}}</ref><ref>{{Cite journal|last=Tong|first=Jingcheng|date=1994|title=Kummer's Test Gives Characterizations for Convergence or Divergence of all Positive Series|url=https://www.jstor.org/stable/2974907|journal=The American Mathematical Monthly|volume=101|issue=5|pages=450–452|doi=10.2307/2974907|jstor=2974907}}</ref><ref>{{Cite journal|last=Samelson|first=Hans|date=1995|title=More on Kummer's Test|url=https://www.tandfonline.com/doi/full/10.1080/00029890.1995.12004667|journal=The American Mathematical Monthly|language=en|volume=102|issue=9|pages=817–818|doi=10.1080/00029890.1995.12004667|issn=0002-9890}}</ref>

(1)  <math> \sum a_n</math> converges if and only if there is a sequence <math>b_{n}</math> of positive numbers and a real number ''c'' > 0 such that <math>b_k (a_{k}/a_{k+1}) - b_{k+1} \ge c</math>.

(2)  <math> \sum a_n</math> diverges if and only if there is a sequence <math>b_{n}</math> of positive numbers such that <math>b_k (a_{k}/a_{k+1}) - b_{k+1} \le 0</math>

and <math> \sum 1/b_{n}</math> diverges.

===Abu-Mostafa's test===
Let <math>\sum_{n=1}^\infty a_n</math> be an infinite series with real terms and let <math>f:\R\to\R</math> be any real function such that <math>f(1/n)=a_n</math> for all positive integers ''n'' and the second derivative <math>f''</math> exists at <math>x=0</math>. Then <math>\sum_{n=1}^\infty a_n</math> converges absolutely if <math>f(0)=f'(0)=0</math> and diverges otherwise.<ref>{{Cite journal|last=Abu-Mostafa|first=Yaser|date=1984|title=A Differentiation Test for Absolute Convergence|url=https://work.caltech.edu/paper/Abu-Mostafa17280.pdf|journal=Mathematics Magazine|volume=57|issue=4|pages=228–231}}</ref>

===Notes===

*For some specific types of series there are more specialized convergence tests, for instance for [[Fourier series]] there is the [[Dini test]].

== Examples ==
Consider the series

{{NumBlk|:|<math>\sum_{n=1}^{\infty} \frac{1}{n^\alpha}.</math>|{{EquationRef|i}}}}

[[Cauchy condensation test]] implies that ({{EquationNote|i}}) is finitely convergent if

{{NumBlk|:|<math> \sum_{n=1}^\infty 2^n \left( \frac 1 {2^n}\right)^\alpha </math>|{{EquationRef|ii}}}}

is finitely convergent. Since

: <math>\sum_{n=1}^\infty 2^n \left( \frac 1 {2^n} \right)^\alpha = \sum_{n=1}^\infty 2^{n-n\alpha} = \sum_{n=1}^\infty 2^{(1-\alpha) n} </math>

({{EquationNote|ii}}) is a geometric series with ratio <math> 2^{(1-\alpha)} </math>. ({{EquationNote|ii}}) is finitely convergent if its ratio is less than one (namely {{nowrap|<math>\alpha > 1</math>).}} Thus, ({{EquationNote|i}}) is finitely convergent [[if and only if]] {{nowrap|<math>\alpha > 1</math>.}}

== Convergence of products ==
While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of [[infinite product]]s. This can be achieved using following theorem: Let <math>\left \{ a_n \right \}_{n=1}^\infty</math> be a sequence of positive numbers. Then the infinite product <math>\prod_{n=1}^\infty (1 + a_n)</math> converges [[if and only if]] the series <math>\sum_{n=1}^\infty a_n</math> converges. Also similarly, if <math>0 < a_n < 1</math> holds, then <math>\prod_{n=1}^\infty (1 - a_n)</math> approaches a non-zero limit if and only if the series <math>\sum_{n=1}^\infty a_n</math> converges .

This can be proved by taking the logarithm of the product and using limit comparison test.<ref>{{cite web|url=https://cornellmath.wordpress.com/2008/01/26/convergence-of-infinite-products/|title=Convergence of Infinite Products|first=Jim|last=Belk|date=26 January 2008}}</ref>

==See also==
*[[L'Hôpital's rule]]
*[[Shift rule]]

== References ==
{{Reflist}}

== Further reading ==
* {{cite book |first=Louis |last=Leithold |author-link=Louis Leithold |title=The Calculus, with Analytic Geometry |location=New York |publisher=Harper & Row |edition=2nd |year=1972 |isbn=0-06-043959-9 |pages=655–737 }}

[[Category:Convergence tests| ]]