ratio test

File:Decision diagram for the ratio test.svg

The usual form of the test makes use of the limit

{{NumBlk|:|$L\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|.$|{{EquationRef|1}}}}

The ratio test states that:

• if L < 1 then the series converges absolutely;
• if L > 1 then the series diverges;
• if L = 1 or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.

It is possible to make the ratio test applicable to certain cases where the limit L fails to exist, if limit superior and limit inferior are used. The test criteria can also be refined so that the test is sometimes conclusive even when L = 1. More specifically, let

:$R\; =\; \backslash lim\backslash sup\; \backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|$

:$r\; =\; \backslash lim\backslash inf\; \backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|$.

Then the ratio test states that:{{harvnb|Rudin|1976|loc=§3.34}}{{harvnb|Apostol|1974|loc=§8.14}}

• if R < 1, the series converges absolutely;
• if r > 1, the series diverges; or equivalently if $\backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|>\; 1$ for all large n (regardless of the value of r), the series also diverges; this is because $|a\_n|$ is nonzero and increasing and hence {{mvar|a_n}} does not approach zero;
• the test is otherwise inconclusive.

If the limit L in ({{EquationNote|1}}) exists, we must have L = R = r. So the original ratio test is a weaker version of the refined one.

Consider the series

:$\backslash sum\_\{n=1\}^\backslash infty\backslash frac\{n\}\{e^n\}$

Applying the ratio test, one computes the limit

:

Since this limit is less than 1, the series converges.

Consider the series

:$\backslash sum\_\{n=1\}^\backslash infty\backslash frac\{e^n\}\{n\}.$

Putting this into the ratio test:

:$L\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; \backslash left|\; \backslash frac\{a\_\{n+1\}\}\{a\_n\}\; \backslash right|\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\; \backslash left|\; \backslash frac\{\backslash frac\{e^\{n+1\}\}\{n+1\}\}\{\backslash frac\{e^n\}\{n\}\}\; \backslash right|$

= e > 1.

Thus the series diverges.

Consider the three series

:$\backslash sum\_\{n=1\}^\backslash infty\; 1,$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{n^2\},$

:$\backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{(-1)^\{n+1\}\}\{n\}.$

The first series (1 + 1 + 1 + 1 + ⋯) diverges, the second (the one central to the Basel problem) converges absolutely and the third (the alternating harmonic series) converges conditionally. However, the term-by-term magnitude ratios $\backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|$ of the three series are $1,$   $\backslash frac\{n^2\}\{(n+1)^2\}$    and   $\backslash frac\{n\}\{n+1\}$. So, in all three, the limit $\backslash lim\_\{n\backslash to\backslash infty\}\backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|$ is equal to 1. This illustrates that when L = 1, the series may converge or diverge: the ratio test is inconclusive. In such cases, more refined tests are required to determine convergence or divergence.

File:Ratio test proof.svg

Below is a proof of the validity of the generalized ratio test.

Suppose that $r=\backslash liminf\_\{n\backslash to\backslash infty\}\backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|>1$. We also suppose that $(a\_n)$ has infinite non-zero members, otherwise the series is just a finite sum hence it converges. Then there exists some $\backslash ell\backslash in(1;r)$ such that there exists a natural number $n\_0\backslash ge2$ satisfying $a\_\{n\_0\}\backslash ne0$ and $\backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|>\backslash ell$ for all $n\backslash ge\; n\_0$, because if no such $\backslash ell$ exists then there exists arbitrarily large $n$ satisfying for every $\backslash ell\backslash in(1;r)$, then we can find a subsequence $\backslash left(a\_\{n\_k\}\backslash right)\_\{k=1\}^\backslash infty$ satisfying

Now suppose . Similar to the above case, we may find a natural number $n\_1$ and a $c\backslash in(R;1)$ such that $|a\_n|\backslash le\; c^\{n-n\_1\}\backslash left|a\_\{n\_1\}\backslash right|$ for $n\backslash ge\; n\_1$. Then

$$\backslash sum\_\{n=1\}^\backslash infty\; |a\_n|=\backslash sum\_\{k=1\}^\{n\_1-1\}|a\_k|+\backslash sum\_\{n=n\_1\}^\backslash infty\; |a\_n|\backslash le\backslash sum\_\{k=1\}^\{n\_1-1\}|a\_k|+\backslash sum\_\{n=n\_1\}^\backslash infty\; c^\{n-n\_1\}|a\_\{n\_1\}|=\backslash sum\_\{k=1\}^\{n\_1-1\}|a\_k|+\backslash left|a\_\{n\_1\}\backslash right|\backslash sum\_\{n=0\}^\backslash infty\; c^n.$$

The series $\backslash sum\_\{n=0\}^\backslash infty\; c^n$ is the geometric series with common ratio $c\backslash in(0;1)$, hence $\backslash sum\_\{n=0\}^\backslash infty\; c^n=\backslash frac\{1\}\{1-c\}$ which is finite. The sum $\backslash sum\_\{k=1\}^\{n\_1-1\}|a\_k|$ is a finite sum and hence it is bounded, this implies the series $\backslash sum\_\{n=1\}^\backslash infty\; |a\_n|$ converges by the monotone convergence theorem and the series $\backslash sum\_\{n=1\}^\backslash infty\; a\_n$ converges by the absolute convergence test.

When the limit $\backslash left|\backslash frac\{a\_\{n+1\}\}\{a\_n\}\backslash right|$ exists and equals to $L$ then $r=R=L$, this gives the original ratio test.

As seen in the previous example, the ratio test may be inconclusive when the limit of the ratio is 1. Extensions to the ratio test, however, sometimes allow one to deal with this case.{{cite book |last=Bromwich |first=T. J. I'A |date=1908 |title=An Introduction To The Theory of Infinite Series |publisher=Merchant Books |author-link= Thomas John I'Anson Bromwich}}{{cite book |last=Knopp |first=Konrad |date=1954 |title=Theory and Application of Infinite Series |url=https://archive.org/details/theoryandapplica031692mbp/page/n5 |location=London |publisher=Blackie & Son Ltd. |author-link=Konrad Knopp }}

{{cite journal |last1=Tong |first1=Jingcheng|date=May 1994|title=Kummer's Test Gives Characterizations for Convergence or Divergence of all Positive Series|jstor=2974907|journal=The American Mathematical Monthly |volume=101 |issue=5 |pages=450–452 |doi=10.2307/2974907 }}

{{cite journal |last1=Ali |first1=Sayel A.|date=2008 |title=The mth Ratio Test: New Convergence Test for Series |url= https://www.tandfonline.com/doi/abs/10.1080/00029890.2008.11920558 |journal=The American Mathematical Monthly |volume=115 |issue=6 |pages=514–524 |doi= 10.1080/00029890.2008.11920558|s2cid=16336333|access-date=4 September 2024 |url-access=subscription }}

{{cite journal |last1=Samelson |first1=Hans|date=November 1995|title=More on Kummer's Test|jstor=2974510|journal=The American Mathematical Monthly |volume=102 |issue=9 |pages=817–818 |doi=10.2307/2974510 }}{{cite web |url= http://sites.math.washington.edu/~morrow/336_12/papers/kyle.pdf|title= The mth Ratio Convergence Test and Other Unconventional Convergence Tests|last= Blackburn|first=Kyle |date=4 May 2012 |publisher=University of Washington College of Arts and Sciences |access-date= 27 November 2018 }}{{cite thesis |last1=Ďuriš |first1=František |date=2009 |title=Infinite series: Convergence tests |type=Bachelor's thesis |publisher=Katedra Informatiky, Fakulta Matematiky, Fyziky a Informatiky, Univerzita Komenského, Bratislava |url= http://www.dcs.fmph.uniba.sk/bakalarky/obhajene/Detail.php?id=90|access-date= 28 November 2018 }}{{cite arXiv |last1=Ďuriš |first1=František |date= 2 February 2018 |title=On Kummer's test of convergence and its relation to basic comparison tests |eprint=1612.05167 |class=math.HO }}

In all the tests below one assumes that Σa_n is a sum with positive a_n. These tests also may be applied to any series with a finite number of negative terms. Any such series may be written as:

:$\backslash sum\_\{n=1\}^\backslash infty\; a\_n\; =\; \backslash sum\_\{n=1\}^N\; a\_n+\backslash sum\_\{n=N+1\}^\backslash infty\; a\_n$

where a_N is the highest-indexed negative term. The first expression on the right is a partial sum which will be finite, and so the convergence of the entire series will be determined by the convergence properties of the second expression on the right, which may be re-indexed to form a series of all positive terms beginning at n=1.

Each test defines a test parameter (ρ_n) which specifies the behavior of that parameter needed to establish convergence or divergence. For each test, a weaker form of the test exists which will instead place restrictions upon lim_n->∞ρ_n.

All of the tests have regions in which they fail to describe the convergence properties of Σa_n. In fact, no convergence test can fully describe the convergence properties of the series. This is because if Σa_n is convergent, a second convergent series Σb_n can be found which converges more slowly: i.e., it has the property that lim_n->∞ (b_n/a_n) = ∞. Furthermore, if Σa_n is divergent, a second divergent series Σb_n can be found which diverges more slowly: i.e., it has the property that lim_n->∞ (b_n/a_n) = 0. Convergence tests essentially use the comparison test on some particular family of a_n, and fail for sequences which converge or diverge more slowly.

Augustus De Morgan proposed a hierarchy of ratio-type tests

The ratio test parameters ($\backslash rho\_n$) below all generally involve terms of the form $D\_n\; a\_n/a\_\{n+1\}-D\_\{n+1\}$. This term may be multiplied by $a\_\{n+1\}/a\_n$ to yield $D\_n-D\_\{n+1\}a\_\{n+1\}/a\_n$. This term can replace the former term in the definition of the test parameters and the conclusions drawn will remain the same. Accordingly, there will be no distinction drawn between references which use one or the other form of the test parameter.

The first test in the De Morgan hierarchy is the ratio test as described above.

This extension is due to Joseph Ludwig Raabe. Define:

:$\backslash rho\_n\; \backslash equiv\; n\backslash left(\backslash frac\{a\_n\}\{a\_\{n+1\}\}-1\backslash right)$

(and some extra terms, see Ali, Blackburn, Feld, Duris (none), Duris2){{Clarify|date=September 2024}}

The series will:

• Converge when there exists a c>1 such that $\backslash rho\_n\; \backslash ge\; c$ for all n>N.
• Diverge when $\backslash rho\_n\; \backslash le\; 1$ for all n>N.
• Otherwise, the test is inconclusive.

For the limit version,{{mathworld|title=Raabe's Test|urlname=RaabesTest}} the series will:

• Converge if $\backslash rho=\backslash lim\_\{n\backslash to\backslash infty\}\backslash rho\_n>1$ (this includes the case ρ = ∞)
• Diverge if .
• If ρ = 1, the test is inconclusive.

When the above limit does not exist, it may be possible to use limits superior and inferior. The series will:

• Converge if $\backslash liminf\_\{n\; \backslash to\; \backslash infty\}\; \backslash rho\_n\; >\; 1$
• Diverge if
• Otherwise, the test is inconclusive.

Defining $\backslash rho\_n\; \backslash equiv\; n\backslash left(\backslash frac\{a\_n\}\{a\_\{n+1\}\}-1\backslash right)$, we need not assume the limit exists; if , then $\backslash sum\; a\_n$ diverges, while if $\backslash liminf\; \backslash rho\_n>1$ the sum converges.

The proof proceeds essentially by comparison with $\backslash sum1/n^R$. Suppose first that . Of course

if then $a\_\{n+1\}\backslash ge\; a\_n$ for large $n$, so the sum diverges; assume then that . There exists such that $\backslash rho\_n\backslash le\; R$ for all $n\backslash ge\; N$, which is to say that $a\_\{n\}/a\_\{n+1\}\backslash le\; \backslash left(1+\backslash frac\; Rn\backslash right)\backslash le\; e^\{R/n\}$. Thus $a\_\{n+1\}\backslash ge\; a\_ne^\{-R/n\}$, which implies that

$a\_\{n+1\}\backslash ge\; a\_Ne^\{-R(1/N+\backslash dots+1/n)\}\backslash ge\; ca\_Ne^\{-R\backslash log(n)\}=ca\_N/n^R$ for $n\backslash ge\; N$; since this shows that $\backslash sum\; a\_n$ diverges.

The proof of the other half is entirely analogous, with most of the inequalities simply reversed. We need a preliminary inequality to use

in place of the simple

$\backslash log\backslash left(1+\backslash frac\; Rn\backslash right)=\backslash frac\; Rn+O\backslash left(\backslash frac\; 1\{n^2\}\backslash right)$. So $\backslash log\backslash left(\backslash left(1+\backslash frac\; RN\backslash right)\backslash dots\backslash left(1+\backslash frac\; Rn\backslash right)\backslash right)$

=R\left(\frac 1N+\dots+\frac 1n\right)+O(1)=R\log(n)+O(1); hence $\backslash left(1+\backslash frac\; RN\backslash right)\backslash dots\backslash left(1+\backslash frac\; Rn\backslash right)\backslash ge\; cn^R$.

Suppose now that $\backslash liminf\backslash rho\_n>1$. Arguing as in the first paragraph, using the inequality established in the previous paragraph, we see that there exists $R>1$ such that $a\_\{n+1\}\backslash le\; ca\_Nn^\{-R\}$ for $n\backslash ge\; N$; since $R>1$ this shows that $\backslash sum\; a\_n$ converges.

This extension is due to Joseph Bertrand and Augustus De Morgan.

Defining:

:$\backslash rho\_n\; \backslash equiv\; n\; \backslash ln\; n\backslash left(\backslash frac\{a\_n\}\{a\_\{n+1\}\}-1\backslash right)-\backslash ln\; n$

Bertrand's test asserts that the series will:

• Converge when there exists a c>1 such that $\backslash rho\_n\; \backslash ge\; c$ for all n>N.
• Diverge when $\backslash rho\_n\; \backslash le\; 1$ for all n>N.
• Otherwise, the test is inconclusive.

For the limit version, the series will:

• Converge if $\backslash rho=\backslash lim\_\{n\backslash to\backslash infty\}\backslash rho\_n>1$ (this includes the case ρ = ∞)
• Diverge if .
• If ρ = 1, the test is inconclusive.

When the above limit does not exist, it may be possible to use limits superior and inferior.{{mathworld|title=Bertrand's Test|urlname=BertrandsTest}} The series will:

• Converge if $\backslash liminf\; \backslash rho\_n\; >\; 1$
• Diverge if
• Otherwise, the test is inconclusive.

This extension probably appeared at the first time by Margaret Martin in 1941.{{cite journal|url=https://www.ams.org/journals/bull/1941-47-06/S0002-9904-1941-07477-X/S0002-9904-1941-07477-X.pdf |last1=Martin |first1=Margaret |date=1941 |title=A sequence of limit tests for the convergence of series |journal=Bulletin of the American Mathematical Society |volume=47 |issue=6|pages=452–457 |doi=10.1090/S0002-9904-1941-07477-X |doi-access=free }} A short proof based on Kummer's test and without technical assumptions (such as existence of the limits, for example) was provided by Vyacheslav Abramov in 2019.{{cite journal|last1=Abramov |first1=Vyacheslav M. |date=May 2020 |title=Extension of the Bertrand–De Morgan test and its application |journal=The American Mathematical Monthly |volume=127 |issue=5 |pages=444–448 |doi=10.1080/00029890.2020.1722551 |arxiv=1901.05843 |s2cid=199552015 }}

Let $K\backslash geq1$ be an integer, and let $\backslash ln\_\{(K)\}(x)$ denote the $K$th iterate of natural logarithm, i.e. $\backslash ln\_\{(1)\}(x)=\backslash ln\; (x)$ and for any $2\backslash leq\; k\backslash leq\; K$,

$\backslash ln\_\{(k)\}(x)=\backslash ln\_\{(k-1)\}(\backslash ln\; (x))$.

Suppose that the ratio $a\_n/a\_\{n+1\}$, when $n$ is large, can be presented in the form

:$\backslash frac\{a\_n\}\{a\_\{n+1\}\}=1+\backslash frac\{1\}\{n\}+\backslash frac\{1\}\{n\}\backslash sum\_\{i=1\}^\{K-1\}\backslash frac\{1\}\{\backslash prod\_\{k=1\}^i\backslash ln\_\{(k)\}(n)\}+\backslash frac\{\backslash rho\_n\}\{n\backslash prod\_\{k=1\}^K\backslash ln\_\{(k)\}(n)\},\; \backslash quad\; K\backslash geq1.$

(The empty sum is assumed to be 0. With $K=1$, the test reduces to Bertrand's test.)

The value $\backslash rho\_\{n\}$ can be presented explicitly in the form

:$\backslash rho\_\{n\}\; =\; n\backslash prod\_\{k=1\}^K\backslash ln\_\{(k)\}(n)\backslash left(\backslash frac\{a\_n\}\{a\_\{n+1\}\}-1\backslash right)-\backslash sum\_\{j=1\}^K\backslash prod\_\{k=1\}^j\backslash ln\_\{(K-k+1)\}(n).$

Extended Bertrand's test asserts that the series

• Converge when there exists a $c>1$ such that $\backslash rho\_n\; \backslash geq\; c$ for all $n>N$.
• Diverge when $\backslash rho\_n\; \backslash leq\; 1$ for all $n>N$.
• Otherwise, the test is inconclusive.

For the limit version, the series

• Converge if $\backslash rho=\backslash lim\_\{n\backslash to\backslash infty\}\backslash rho\_n>1$ (this includes the case $\backslash rho\; =\; \backslash infty$)
• Diverge if .
• If $\backslash rho\; =\; 1$, the test is inconclusive.

When the above limit does not exist, it may be possible to use limits superior and inferior. The series

• Converge if $\backslash liminf\; \backslash rho\_n\; >\; 1$
• Diverge if
• Otherwise, the test is inconclusive.

For applications of Extended Bertrand's test see birth–death process.

This extension is due to Carl Friedrich Gauss.

Assuming a_n > 0 and r > 1, if a bounded sequence C_n can be found such that for all n:

:$\backslash frac\{a\_n\}\{a\_\{n+1\}\}=1+\backslash frac\{\backslash rho\}\{n\}+\backslash frac\{C\_n\}\{n^r\}$

then the series will:

• Converge if $\backslash rho>1$
• Diverge if $\backslash rho\; \backslash le\; 1$

This extension is due to Ernst Kummer.

Let ζ_n be an auxiliary sequence of positive constants. Define

:$\backslash rho\_n\; \backslash equiv\; \backslash left(\backslash zeta\_n\; \backslash frac\{a\_n\}\{a\_\{n+1\}\}\; -\; \backslash zeta\_\{n+1\}\backslash right)$

Kummer's test states that the series will:

• Converge if there exists a $c>0$ such that $\backslash rho\_n\; \backslash ge\; c$ for all n>N. (Note this is not the same as saying $\backslash rho\_n\; >\; 0$)
• Diverge if $\backslash rho\_n\; \backslash le\; 0$ for all n>N and $\backslash sum\_\{n=1\}^\backslash infty\; 1/\backslash zeta\_n$ diverges.

For the limit version, the series will:{{mathworld|title=Kummer's Test|urlname=KummersTest}}

• Converge if $\backslash lim\_\{n\backslash to\backslash infty\}\backslash rho\_n>0$ (this includes the case ρ = ∞)
• Diverge if and $\backslash sum\_\{n=1\}^\backslash infty\; 1/\backslash zeta\_n$ diverges.
• Otherwise the test is inconclusive

When the above limit does not exist, it may be possible to use limits superior and inferior. The series will

• Converge if $\backslash liminf\_\{n\; \backslash to\; \backslash infty\}\; \backslash rho\_n\; >0$
• Diverge if and $\backslash sum\; 1/\backslash zeta\_n$ diverges.

All of the tests in De Morgan's hierarchy except Gauss's test can easily be seen as special cases of Kummer's test:

• For the ratio test, let ζ_n=1. Then:

::$\backslash rho\_\backslash text\{Kummer\}\; =\; \backslash left(\backslash frac\{a\_n\}\{a\_\{n+1\}\}-1\backslash right)\; =\; 1/\backslash rho\_\backslash text\{Ratio\}-1$

• For Raabe's test, let ζ_n=n. Then:

::$\backslash rho\_\backslash text\{Kummer\}\; =\; \backslash left(n\backslash frac\{a\_n\}\{a\_\{n+1\}\}-(n+1)\backslash right)\; =\; \backslash rho\_\backslash text\{Raabe\}-1$

• For Bertrand's test, let ζ_n=n ln(n). Then:

::$\backslash rho\_\backslash text\{Kummer\}\; =\; n\; \backslash ln(n)\backslash left(\backslash frac\{a\_n\}\{a\_\{n+1\}\}\backslash right)-(n+1)\backslash ln(n+1)$

:Using $\backslash ln(n+1)=\backslash ln(n)+\backslash ln(1+1/n)$ and approximating $\backslash ln(1+1/n)\backslash rightarrow\; 1/n$ for large n, which is negligible compared to the other terms, $\backslash rho\_\backslash text\{Kummer\}$ may be written:

::$\backslash rho\_\backslash text\{Kummer\}\; =\; n\; \backslash ln(n)\backslash left(\backslash frac\{a\_n\}\{a\_\{n+1\}\}-1\backslash right)-\backslash ln(n)-1\; =\; \backslash rho\_\backslash text\{Bertrand\}-1$

• For Extended Bertrand's test, let $\backslash zeta\_n=n\backslash prod\_\{k=1\}^K\backslash ln\_\{(k)\}(n).$ From the Taylor series expansion for large $n$ we arrive at the approximation

::$\backslash ln\_\{(k)\}(n+1)=\backslash ln\_\{(k)\}(n)+\backslash frac\{1\}\{n\backslash prod\_\{j=1\}^\{k-1\}\backslash ln\_\{(j)\}(n)\}+O\backslash left(\backslash frac\{1\}\{n^2\}\backslash right),$

where the empty product is assumed to be 1. Then,

::$\backslash rho\_\backslash text\{Kummer\}\; =\; n\backslash prod\_\{k=1\}^K\backslash ln\_\{(k)\}(n)\backslash frac\{a\_n\}\{a\_\{n+1\}\}-(n+1)\backslash left[\backslash prod\_\{k=1\}^K\backslash left(\backslash ln\_\{(k)\}(n)+\backslash frac\{1\}\{n\backslash prod\_\{j=1\}^\{k-1\}\backslash ln\_\{(j)\}(n)\}\backslash right)\backslash right]+o(1)$

=n\prod_{k=1}^K\ln_{(k)}(n)\left(\frac{a_n}{a_{n+1}}-1\right)-\sum_{j=1}^K\prod_{k=1}^j\ln_{(K-k+1)}(n)-1+o(1).

Hence,

::$\backslash rho\_\backslash text\{Kummer\}\; =\; \backslash rho\_\backslash text\{Extended\; Bertrand\}-1.$

Note that for these four tests, the higher they are in the De Morgan hierarchy, the more slowly the $1/\backslash zeta\_n$ series diverges.

If $\backslash rho\_n>0$ then fix a positive number . There exists

a natural number $N$ such that for every $n>N,$

:$\backslash delta\backslash leq\backslash zeta\_\{n\}\backslash frac\{a\_\{n\}\}\{a\_\{n+1\}\}-\backslash zeta\_\{n+1\}.$

Since $a\_\{n+1\}>0$, for every $n>\; N,$

:$0\backslash leq\; \backslash delta\; a\_\{n+1\}\backslash leq\; \backslash zeta\_\{n\}a\_\{n\}-\backslash zeta\_\{n+1\}a\_\{n+1\}.$

In particular $\backslash zeta\_\{n+1\}a\_\{n+1\}\backslash leq\; \backslash zeta\_\{n\}a\_\{n\}$ for all $n\backslash geq\; N$ which means that starting from the index

$N$

the sequence $\backslash zeta\_\{n\}a\_\{n\}>0$ is monotonically decreasing and

positive which in particular implies that it is bounded below by 0. Therefore, the limit

:$\backslash lim\_\{n\backslash to\backslash infty\}\backslash zeta\_\{n\}a\_\{n\}=L$ exists.

This implies that the positive telescoping series

:$\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash left(\backslash zeta\_\{n\}a\_\{n\}-\backslash zeta\_\{n+1\}a\_\{n+1\}\backslash right)$ is convergent,

and since for all $n>N,$

:$\backslash delta\; a\_\{n+1\}\backslash leq\; \backslash zeta\_\{n\}a\_\{n\}-\backslash zeta\_\{n+1\}a\_\{n+1\}$

by the direct comparison test for positive series, the series

$\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash delta\; a\_\{n+1\}$ is convergent.

On the other hand, if , then there is an N such that $\backslash zeta\_n\; a\_n$ is increasing for $n>N$. In particular, there exists an $\backslash epsilon>0$ for which $\backslash zeta\_n\; a\_n>\backslash epsilon$ for all $n>N$, and so $\backslash sum\_n\; a\_n=\backslash sum\_n\; \backslash frac\{a\_n\backslash zeta\_n\}\{\backslash zeta\_n\}$ diverges by comparison with $\backslash sum\_n\; \backslash frac\; \backslash epsilon\; \{\backslash zeta\_n\}$.

A new version of Kummer's test was established by Tong. See also {{cite arXiv |last1=Abramov |first1=Vyacheslav, M. |date= 21 June 2021 |title=A simple proof of Tong's theorem |eprint=2106.13808 |class=math.HO }}

for further discussions and new proofs. The provided modification of Kummer's theorem characterizes

all positive series, and the convergence or divergence can be formulated in the form of two necessary and sufficient conditions, one for convergence and another for divergence.

• Series $\backslash sum\_\{n=1\}^\backslash infty\; a\_n$ converges if and only if there exists a positive sequence $\backslash zeta\_n$, $n=1,2,\backslash dots$, such that $\backslash zeta\_n\backslash frac\{a\_n\}\{a\_\{n+1\}\}-\backslash zeta\_\{n+1\}\backslash geq\; c>0.$

• Series $\backslash sum\_\{n=1\}^\backslash infty\; a\_n$ diverges if and only if there exists a positive sequence $\backslash zeta\_n$, $n=1,2,\backslash dots$, such that $\backslash zeta\_n\backslash frac\{a\_n\}\{a\_\{n+1\}\}-\backslash zeta\_\{n+1\}\backslash leq0,$ and $\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{1\}\{\backslash zeta\_n\}=\backslash infty.$

The first of these statements can be simplified as follows:

{{cite journal |last1=Abramov |first1=Vyacheslav M.|date=May 2022|title=Evaluating the sum of convergent positive series|url=http://elib.mi.sanu.ac.rs/files/journals/publ/131/publn131p41-53.pdf | journal=Publications de l'Institut Mathématique |series=Nouvelle Série |volume=111 |issue=125 |pages=41–53 |doi=10.2298/PIM2225041A|s2cid=237499616 }}

• Series $\backslash sum\_\{n=1\}^\backslash infty\; a\_n$ converges if and only if there exists a positive sequence $\backslash zeta\_n$, $n=1,2,\backslash dots$, such that $\backslash zeta\_n\backslash frac\{a\_n\}\{a\_\{n+1\}\}-\backslash zeta\_\{n+1\}=1.$

The second statement can be simplified similarly:

• Series $\backslash sum\_\{n=1\}^\backslash infty\; a\_n$ diverges if and only if there exists a positive sequence $\backslash zeta\_n$, $n=1,2,\backslash dots$, such that $\backslash zeta\_n\backslash frac\{a\_n\}\{a\_\{n+1\}\}-\backslash zeta\_\{n+1\}=0,$ and $\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{1\}\{\backslash zeta\_n\}=\backslash infty.$

However, it becomes useless, since the condition $\backslash sum\_\{n=1\}^\{\backslash infty\}\backslash frac\{1\}\{\backslash zeta\_n\}=\backslash infty$ in this case reduces to the original claim $\backslash sum\_\{n=1\}^\{\backslash infty\}a\_n=\backslash infty.$

Another ratio test that can be set in the framework of Kummer's theorem was presented by Orrin Frink{{cite journal |last1=Frink |first1=Orrin |date=October 1948|title=A ratio test |journal=Bulletin of the American Mathematical Society |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-54/issue-10/A-ratio-test/bams/1183512386.full |volume=54 |issue=10 |pages=953-953}} 1948.

Suppose $a\_n$ is a sequence in $\backslash mathbb\{C\}\backslash setminus\backslash \{0\backslash \}$,

• If $\backslash limsup\_\{n\backslash rightarrow\backslash infty\}\backslash Big(\backslash frac$

  a_{n+1}

  a_n

  \Big)^n<\frac1e , then the series $\backslash sum\_na\_n$ converges absolutely.
• If there is $N\backslash in\backslash mathbb\{N\}$ such that $\backslash Big(\backslash frac$

  a_{n+1}

  a_n

  \Big)^n\geq\frac1e for all $n\backslash geq\; N$, then $\backslash sum\_n|a\_n|$ diverges.

This result reduces to a comparison of $\backslash sum\_n|a\_n|$ with a power series $\backslash sum\_n\; n^\{-p\}$, and can be seen to be related to Raabe's test.{{cite journal |date=1949|last1=Stark |first1=Marceli |title=On the ratio test of Frink |journal=Colloquium Mathematicum | volume=2 |issue=1 |pages=46-47 }}

A more refined ratio test is the second ratio test:

For $a\_n>0$ define:

By the second ratio test, the series will:

• Converge if
• Diverge if $L>\backslash frac\{1\}\{2\}$
• If $L=\backslash frac\{1\}\{2\}$ then the test is inconclusive.

If the above limits do not exist, it may be possible to use the limits superior and inferior. Define:

Then the series will:

• Converge if
• Diverge if $\backslash ell>\backslash frac\{1\}\{2\}$
• If $\backslash ell\; \backslash le\; \backslash frac\{1\}\{2\}\; \backslash le\; L$ then the test is inconclusive.

This test is a direct extension of the second ratio test. For $0\backslash leq\; k\backslash leq\; m-1,$ and positive $a\_n$ define:

By the $m$th ratio test, the series will:

• Converge if
• Diverge if $L>\backslash frac\{1\}\{m\}$
• If $L=\backslash frac\{1\}\{m\}$ then the test is inconclusive.

If the above limits do not exist, it may be possible to use the limits superior and inferior. For $0\backslash leq\; k\backslash leq\; m-1$ define:

Then the series will:

• Converge if
• Diverge if $\backslash ell>\backslash frac\{1\}\{m\}$
• If $\backslash ell\; \backslash leq\; \backslash frac\{1\}\{m\}\; \backslash leq\; L$, then the test is inconclusive.

This test is an extension of the $m$th ratio test.{{cite journal|url=https://www.ems-ph.org/journals/show_abstract.php?issn=0013-6018&vol=67&iss=4&rank=2|last1=Ali |first1=Sayel |last2=Cohen |first2=Marion Deutsche |date=2012 |title=phi-ratio tests |journal=Elemente der Mathematik |volume=67 |issue= 4|pages=164–168 |doi=10.4171/EM/206 |doi-access=free }}

Assume that the sequence $a\_n$ is a positive decreasing sequence.

Let $\backslash varphi:\backslash mathbb\{Z\}^+\backslash to\backslash mathbb\{Z\}^+$ be such that $\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{n\}\{\backslash varphi(n)\}$ exists. Denote $\backslash alpha=\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{n\}\{\backslash varphi(n)\}$, and assume .

Assume also that $\backslash lim\_\{n\backslash to\backslash infty\}\backslash frac\{a\_\{\backslash varphi(n)\}\}\{a\_n\}=L.$

Then the series will:

• Converge if
• Diverge if $L>\backslash alpha$
• If $L=\backslash alpha$, then the test is inconclusive.

• Root test
• Radius of convergence

{{reflist}}

• {{citation | last1=d'Alembert| first1=J.|author-link=Jean le Rond d'Alembert| year=1768 | title=Opuscules |volume=V|pages=171–183| url=http://gallica.bnf.fr/ark:/12148/bpt6k62424s.image.f192}}.
• {{Citation | last1=Apostol | first1=Tom M. | author1-link=Tom M. Apostol | title=Mathematical analysis | publisher=Addison-Wesley | edition=2nd | isbn=978-0-201-00288-1 | year=1974}}: §8.14.
• {{citation|last=Knopp|first=Konrad|title=Infinite Sequences and Series|location=New York |publisher=Dover Publications|year=1956|isbn=978-0-486-60153-3|bibcode=1956iss..book.....K}}: §3.3, 5.4.
• {{citation|last=Rudin|first=Walter|author-link=Walter Rudin|title=Principles of Mathematical Analysis|edition=3rd|publisher=McGraw-Hill, Inc.|location=New York|year=1976|isbn=978-0-07-054235-8}}: §3.34.
• {{springer|title=Bertrand criterion|id=p/b015780}}
• {{springer|title=Gauss criterion|id=p/g043420}}
• {{springer|title=Kummer criterion|id=p/k055950}}
• {{citation|last2=Whittaker|first2=E. T.|last1=Watson|first1=G. N.|title=A Course in Modern Analysis|edition=4th|publisher=Cambridge University Press|year=1963|isbn=978-0-521-58807-2}}: §2.36, 2.37.

{{Calculus topics}}

Category:Convergence tests

Category:Articles containing proofs

it:Criteri di convergenza#Criterio del rapporto (o di d'Alembert)