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Raabe is Kummer with $\zeta_k = k+1$ : for positive terms, $\liminf\, (k+1)(a_k/a_{k+1} - 1) > 1$ gives convergence and $\limsup < 1$ gives divergence

Statement

Let $(a_k)$ be a sequence of reals with $a_k > 0$ for every $k \in \mathbb{N}$ . Write $k+1$ for the canonical natural $\iota(k+1) \in \mathbb{R}$ , which is positive (Canonical naturals are positive and strictly increasing), take the weights $\zeta_k := k+1$ in Kummer: for positive terms $a_k$ and weights $\zeta_k > 0$ , $\liminf(\zeta_k a_k/a_{k+1} - \zeta_{k+1}) > 0$ gives convergence, and if $\sum 1/\zeta_k$ diverges while that expression is eventually $\le 0$ the series diverges, and put

$R_k \;:=\; (k+1)\left(\frac{a_k}{a_{k+1}} - 1\right) \qquad (k \in \mathbb{N}),$

so that Kummer's expression for these weights is $K_k = R_k - 1$ . Then:

if $\liminf_{k} R_k > 1$ then $\sum a_k$ converges;
if $\limsup_{k} R_k < 1$ then $\sum a_k$ diverges.

The weights are $k+1$ rather than $k$ because $\zeta_0$ has to be positive and $\mathbb{N}$ contains $0$ ; the classical statement, indexed from $1$ , is the same criterion read along the shift $k = j+1$ .

Nothing is claimed when $\liminf_k R_k \le 1 \le \limsup_k R_k$ . The Gauss test proved next is exactly the tool for the borderline case $R_k \to 1$ , where Raabe's test is silent.

Facts & Assumptions

Given: A sequence $(a_k)$ of reals with $a_k > 0$ for every $k$ ; the weights $\zeta_k = \iota(k+1)$ ; and $R_k = (k+1)(a_k/a_{k+1} - 1)$ (Limit superior and limit inferior of a real sequence as $\inf_n \sup_{k \ge n} x_k$ and $\sup_n \inf_{k \ge n} x_k$ in $\overline{\mathbb{R}}$ , Canonical naturals are positive and strictly increasing).

[L1]

Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound there, and $\limsup_k x_k = \inf\{s_n\}$ , $\liminf_k x_k = \sup\{i_n\}$ for the tail bounds $s_n, i_n$ , both existing for every sequence (Every subset of $\overline{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\overline{\mathbb{R}}$ , agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$ , The extended real line $\overline{\mathbb{R}} = \mathbb{R} \cup \{-\infty, +\infty\}$ , its order, and the arithmetic that is left undefined, Limit superior and limit inferior of a real sequence as $\inf_n \sup_{k \ge n} x_k$ and $\sup_n \inf_{k \ge n} x_k$ in $\overline{\mathbb{R}}$ , The tail suprema of any real sequence are nonincreasing in $\overline{\mathbb{R}}$ , 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 $\zeta_k > 0$ , $\liminf(\zeta_k a_k/a_{k+1} - \zeta_{k+1}) > 0$ gives convergence, and if $\sum 1/\zeta_k$ diverges while that expression is eventually $\le 0$ the series diverges).

[L3]

$\sum_{k \ge 1} 1/k^{p}$ converges if and only if $p > 1$ ; at $p = 1$ it therefore diverges (For rational $p > 0$ , $\sum 1/k^p$ converges iff $p > 1$ ). Moreover $k^{1} = \iota(k)$ , the rational power at exponent $1$ being the element itself (Rational powers $a^r$ of a positive base, Existence and uniqueness of $n$ -th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$ , Integer powers $a^m$ ).

[L4]

The canonical naturals $\iota(k+1)$ are positive, and $\iota(k+2) = \iota(k+1) + 1$ (Canonical naturals are positive and strictly increasing).

[L5]

The series $\sum_{k \ge 1} x_k$ from the starting index $1$ is by definition the series of the sequence $j \mapsto x_{j+1}$ (Series, partial sums, convergence and the sum, divergence, and the tail series).

Proof

technique · direct

1.1

The weights $\zeta_k = \iota(k+1)$ are positive for every $k \in \mathbb{N}$ , and the terms $a_k$ are positive, so Kummer's test applies with these data.

givenL4L2

1.2

Suppose $\liminf_k R_k > 1$ . The real $1$ is not an upper bound of the set of tail infima of $(R_k)$ , so there is $N$ with $i_N > 1$ , and $i_N$ is real because $i_N \le R_N$ .

givenL1choose

1.3

Suppose instead $\limsup_k R_k < 1$ . The real $1$ is not a lower bound of the set of tail suprema of $(R_k)$ , so there is $N$ with $s_N < 1$ , and then $R_k \le s_N < 1$ for every $k \ge N$ .

givenL1choose

2.1

Kummer's expression for these weights is $K_k = (k+1)\dfrac{a_k}{a_{k+1}} - (k+2) = (k+1)\left(\dfrac{a_k}{a_{k+1}} - 1\right) - 1 = R_k - 1$ .

step 1.1L4algebra

2.2

The weight series is $\sum_k 1/\zeta_k = \sum_k 1/\iota(k+1)$ , which is precisely the series $\sum_{k \ge 1} 1/k$ from the starting index $1$ , and that is the case $p = 1$ of the $p$ -series, hence divergent.

step 1.1L3L5

3.1

Put $c := i_N - 1 > 0$ . For every $k \ge N$ we have $R_k \ge i_N$ , hence $K_k = R_k - 1 \ge i_N - 1 = c$ .

step 1.2step 2.1L1algebra

3.2

Hence $K_k = R_k - 1 < 0$ , in particular $K_k \le 0$ , for every $k \ge N$ .

step 1.3step 2.1algebra

4.1

So $c$ is a lower bound of $\{K_k : k \ge N\}$ , whence $\liminf_k K_k \ge c > 0$ , and Kummer's convergence criterion gives convergence of $\sum a_k$ , which is claim 1.

step 3.1step 1.1L1L2

5.1

Together with the divergence of the weight series, Kummer's divergence criterion gives divergence of $\sum a_k$ , which is claim 2.

step 3.2step 2.2step 1.1L2∎

Remarks

Raabe's test is a genuine strengthening of the ratio test. Whenever the ratios $a_{k+1}/a_k$ converge to $1$ the ratio test is silent, while $R_k$ may still be bounded away from $1$ on either side; the companion page carries a series with ratio limit exactly $1$ that Raabe decides. The reason is visible in the weights: the divergent comparison series behind the test has moved from $\sum 1$ to the harmonic series, which diverges far more slowly.
The threshold is $1$ and not $0$ , and step 2.1 says why. Kummer's criterion is a statement about $K_k = R_k - 1$ ; the shift by $1$ between the two expressions is the whole difference between the two thresholds, and it comes from $\zeta_{k+1} - \zeta_k = 1$ for these weights.

Depends on

Used by

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Sources

Raabe's 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)

Raabe is Kummer with ζk=k+1\zeta_k = k+1ζk​=k+1: for positive terms, lim inf⁡ (k+1)(ak/ak+1−1)>1\liminf\, (k+1)(a_k/a_{k+1} - 1) > 1liminf(k+1)(ak​/ak+1​−1)>1 gives convergence and lim sup⁡<1\limsup < 1limsup<1 gives divergence