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Kummer with $ζ_{k} = 1$ recovers the ratio test

Statement

Let $(a_{k})$ be a sequence of reals with $a_{k} > 0$ for every $k \in N$ , and put $q_{k} : = a_{k + 1} / a_{k}$ , which is the ratio family of Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence since $∣ a_{k} ∣ = a_{k}$ here. Take the constant weights $ζ_{k} : = 1$ , so that Kummer's expression (Kummer: for positive terms $a_{k}$ and weights $ζ_{k} > 0$ , $lim inf (ζ_{k} a_{k} / a_{k + 1} - ζ_{k + 1}) > 0$ gives convergence, and if $\sum 1 / ζ_{k}$ diverges while that expression is eventually $\leq 0$ the series diverges) is

$K_{k} = \frac{a_{k}}{a_{k + 1}} - 1 (k \in N) .$

Then:

if ${lim sup}_{k} q_{k} < 1$ then ${lim inf}_{k} K_{k} > 0$ , so Kummer's convergence criterion applies and yields convergence of $\sum a_{k}$ ;
if ${lim inf}_{k} q_{k} > 1$ then $\sum 1 / ζ_{k}$ diverges and $K_{k} \leq 0$ from some index on, so Kummer's divergence criterion applies and yields divergence of $\sum a_{k}$ .

Both conclusions are exactly those of Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence for a sequence of positive terms. So the ratio test is the constant-weight case of Kummer's test, and every strengthening of Kummer's test by a better choice of weights is a strengthening of the ratio test.

Facts & Assumptions

Given: A sequence $(a_{k})$ of reals with $a_{k} > 0$ for every $k$ ; the ratios $q_{k} = a_{k + 1} / a_{k}$ , which are positive; the constant weights $ζ_{k} = 1$ ; and Kummer's expression $K_{k} = a_{k} / a_{k + 1} - 1$ (Kummer: for positive terms $a_{k}$ and weights $ζ_{k} > 0$ , $lim inf (ζ_{k} a_{k} / a_{k + 1} - ζ_{k + 1}) > 0$ gives convergence, and if $\sum 1 / ζ_{k}$ diverges while that expression is eventually $\leq 0$ the series diverges, Limit superior and limit inferior of a real sequence as $\inf_{n} \sup_{k \geq n} x_{k}$ and $\sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ ).

[L1]

Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound there (Every subset of $\overline{R}$ has a least upper bound and a greatest lower bound in $\overline{R}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $R$ , The extended real line $\overline{R} = R \cup {- \infty, + \infty}$ , its order, and the arithmetic that is left undefined); ${lim sup}_{k} x_{k} = \inf {s_{n}}$ and ${lim inf}_{k} x_{k} = \sup {i_{n}}$ with $s_{n}, i_{n}$ the tail bounds, and both exist for every sequence (Limit superior and limit inferior of a real sequence as $\inf_{n} \sup_{k \geq n} x_{k}$ and $\sup_{n} \inf_{k \geq n} x_{k}$ in $\overline{R}$ , The tail suprema of any real sequence are nonincreasing in $\overline{R}$ , so the limit superior exists for every sequence).

[L2]

Kummer's test, in both halves, for positive terms and positive weights (Kummer: for positive terms $a_{k}$ and weights $ζ_{k} > 0$ , $lim inf (ζ_{k} a_{k} / a_{k + 1} - ζ_{k + 1}) > 0$ gives convergence, and if $\sum 1 / ζ_{k}$ diverges while that expression is eventually $\leq 0$ the series diverges).

[L3]

Reciprocation on the positives: $0 < x < y$ implies $0 < 1 / y < 1 / x$ (Inverses of positives are positive, and reciprocation reverses order).

[L4]

A series whose terms do not tend to $0$ diverges (If a series converges then its terms tend to $0$ ).

[L5]

For positive terms $∣ a_{k} ∣ = a_{k}$ , so the ratio family of Ratio test: $lim sup ∣ a_{k + 1} / a_{k} ∣ < 1$ gives absolute convergence and hence convergence, and $lim inf ∣ a_{k + 1} / a_{k} ∣ > 1$ gives divergence is the family $q_{k}$ above (Basic properties of the absolute value).

Proof

technique · direct

1.1

The constant weights $ζ_{k} = 1$ are positive, and the terms $a_{k}$ are positive, so Kummer's test applies with these data and its expression is $K_{k} = a_{k} / a_{k + 1} - 1$ .

givenL2

1.2

Each $q_{k}$ is positive, so ${lim sup}_{k} q_{k} \geq 0$ , every tail supremum being at least $q_{n} > 0$ .

givenL1

1.3

Suppose instead ${lim inf}_{k} q_{k} > 1$ . The real $1$ is not an upper bound of ${i_{n}^{'}}$ , the tail infima of $(q_{k})$ , so there is $N$ with $i_{N}^{'} > 1$ , and then $q_{k} \geq i_{N}^{'} > 1$ for every $k \geq N$ .

givenL1choose

1.4

The weight series $\sum 1 / ζ_{k}$ is $\sum 1$ , whose terms are constantly $1$ and so do not tend to $0$ ; it diverges.

givenL4

2.1

Suppose ${lim sup}_{k} q_{k} < 1$ . Then $Λ : = {lim sup}_{k} q_{k}$ lies between the reals $0$ and $1$ and is therefore real; put $t : = (Λ + 1) / 2$ , so that $Λ < t < 1$ and $t \geq 1 / 2 > 0$ .

step 1.2L1choose

2.2

For $k \geq N$ : $a_{k + 1} / a_{k} > 1$ gives $a_{k + 1} > a_{k} > 0$ , hence $a_{k} / a_{k + 1} < 1$ , hence $K_{k} < 0$ and in particular $K_{k} \leq 0$ .

step 1.3L3algebra

3.1

Since $t > \inf {s_{n}}$ , the real $t$ is not a lower bound of ${s_{n}}$ , so there is $N$ with $s_{N} < t$ , and then $q_{k} \leq s_{N} < t$ for every $k \geq N$ .

step 2.1L1choose

3.2

Kummer's divergence criterion therefore applies and $\sum a_{k}$ diverges, which is claim 2.

step 2.2step 1.4step 1.1L2

4.1

For $k \geq N$ : $0 < a_{k + 1} / a_{k} < t$ , so $a_{k} / a_{k + 1} > 1 / t$ , and hence $K_{k} = a_{k} / a_{k + 1} - 1 > 1 / t - 1 = : c$ , where $c > 0$ because $0 < t < 1$ gives $1 / t > 1$ .

step 3.1L3algebra

5.1

So $c$ is a lower bound of ${K_{k} : k \geq N}$ , whence $i_{N} \geq c$ and ${lim inf}_{k} K_{k} = \sup {i_{n}} \geq i_{N} \geq c > 0$ .

step 4.1L1

6.1

Kummer's convergence criterion therefore applies and $\sum a_{k}$ converges, which is claim 1.

step 5.1step 1.1L2

7.1

The hypotheses in claims 1 and 2 are precisely those of the two halves of the ratio test for this sequence, and the conclusions agree, so the ratio test for positive terms is the case $ζ_{k} = 1$ of Kummer's test.

step 6.1step 3.2L5∎

Remarks

What this corollary is for. It is not a new criterion. It fixes the place of the ratio test inside the Kummer family, so that the later choices of weights on this page can be read as improvements on a known test rather than as unrelated criteria.
The ratio test proved earlier is more general in one respect. It allows terms of either sign, provided none vanishes, and concludes convergence of $\sum ∣ a_{k} ∣$ . Kummer's test needs positivity throughout, so the identification above is between the positive-term case of the ratio test and the constant-weight case of Kummer's test, and it says nothing about signed terms.

Depends on

Used by

How the nonnegative tests are ordered by strength, and which of them this page cannot state without the logarithm Remark

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46 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Ratio test (Wikipedia) (standard reference, not scraped)
K. Knopp, Theory and Application of Infinite Series, Ch. IX (standard reference, not scraped)
Thomson, Bruckner, and Bruckner, Elementary Real Analysis (standard reference, not scraped)
Binghamton University notes on Kummer, Raabe, and Gauss tests (standard reference, not scraped)

Kummer with ζk=1ζk​=1 recovers the ratio test