CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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$a_{k} = (- 1)^{k}$ , $b_{k} = k$ have $a_{k} / b_{k} \to 0$ while the difference quotient oscillates, so Stolz-Cesaro has no converse

Statement refuted

Refuted claim: the converse of Stolz-Cesaro, $\infty / \infty$ form: if $b_{k}$ is strictly increasing and unbounded and $(a_{k + 1} - a_{k}) / (b_{k + 1} - b_{k}) \to L$ then $a_{k} / b_{k} \to L$ . That is: if $(b_{k})$ is strictly increasing with range not bounded above and the quotients $a_{k} / b_{k}$ converge, then the difference quotients $(a_{k + 1} - a_{k}) / (b_{k + 1} - b_{k})$ converge too.

The witness is $a_{k} = s_{k}$ , the alternating sequence of The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $N$ their disjoint union, and the unique $(s_{k})$ with $s_{0} = 1$ , $s_{σ (k)} = - s_{k}$ , which satisfies $∣ s_{k} ∣ = 1$ , $s \circ e \equiv 1$ and $s \circ o \equiv - 1$ usually written $(- 1)^{k}$ , and $b_{k} = k$ . Then $b_{0} = 0$ , so the quotient $a_{k} / b_{k}$ is formed for $k \geq 1$ only, exactly as Stolz-Cesaro, $\infty / \infty$ form: if $b_{k}$ is strictly increasing and unbounded and $(a_{k + 1} - a_{k}) / (b_{k + 1} - b_{k}) \to L$ then $a_{k} / b_{k} \to L$ is stated; and there

$\frac{a_{k}}{b_{k}} = \frac{s_{k}}{k}, ∣ \frac{a_{k}}{b_{k}} ∣ = \frac{1}{k} ⟶ 0,$

while the difference quotient is

$\frac{a_{k + 1} - a_{k}}{b_{k + 1} - b_{k}} = s_{k + 1} - s_{k} = - 2 s_{k},$

which takes the value $- 2$ at even $k$ and $2$ at odd $k$ and does not converge.

Facts & Assumptions

Given: The alternating sequence $(s_{k})$ with $s_{0} = 1$ and $s_{σ (k)} = - s_{k}$ ; the sequences $a_{k} : = s_{k}$ and $b_{k} : = k \cdot 1_{R}$ ; and the difference quotients $d_{k} : = (a_{k + 1} - a_{k}) (b_{k + 1} - b_{k})^{- 1}$ .

[L1]

The alternating sequence: $∣ s_{k} ∣ = 1$ for every $k$ , $s_{k + 1} = - s_{k}$ , $s_{e_{j}} = 1$ and $s_{o_{j}} = - 1$ along the even and odd index maps (The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $N$ their disjoint union, and the unique $(s_{k})$ with $s_{0} = 1$ , $s_{σ (k)} = - s_{k}$ , which satisfies $∣ s_{k} ∣ = 1$ , $s \circ e \equiv 1$ and $s \circ o \equiv - 1$ ).

[L2]

$(s_{k})$ is bounded and does not converge (FALSE: every bounded sequence converges).

[L3]

The canonical naturals of $R$ are positive for $k \geq 1$ and strictly increasing (Canonical naturals are positive and strictly increasing), and no real bounds them all (Every complete ordered field is Archimedean); strict monotonicity and boundedness of real sequences (Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences, Lower bound, bounded below, bounded set, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

[L4]

Convergence of real sequences and uniqueness of limits (Limits and Cauchy sequences of reals, A sequence has at most one limit); convergence depends only on a tail (Convergence depends only on the tail); the reciprocal Archimedean property (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L5]

Algebra of limits, in particular that a scalar multiple of a convergent sequence converges (Algebra of limits: sums, scalar multiples, products and quotients).

[L6]

Order arithmetic: $∣ u ∣ \geq 0$ and $∣ u v ∣ = ∣ u ∣ ∣ v ∣$ (Basic properties of the absolute value); a positive element is invertible with positive inverse and reciprocation reverses the order (Inverses of positives are positive, and reciprocation reverses order); the order is total and transitive (Complete ordered field (least-upper-bound property), Ordered field).

[L7]

The hypotheses and conclusion of Stolz-Cesaro in the $\infty / \infty$ form (Stolz-Cesaro, $\infty / \infty$ form: if $b_{k}$ is strictly increasing and unbounded and $(a_{k + 1} - a_{k}) / (b_{k + 1} - b_{k}) \to L$ then $a_{k} / b_{k} \to L$ ).

Counterexample

technique · direct

1.1

$(b_{k})$ is strictly increasing, its range is not bounded above, $b_{0} = 0$ and $b_{k} > 0$ for $k \geq 1$ ; so the hypotheses of Stolz-Cesaro, $\infty / \infty$ form: if $b_{k}$ is strictly increasing and unbounded and $(a_{k + 1} - a_{k}) / (b_{k + 1} - b_{k}) \to L$ then $a_{k} / b_{k} \to L$ on $(b_{k})$ hold with $K_{0} = 1$ .

L3L7

1.2

$∣ s_{k} ∣ = 1$ for every $k$ , and $(s_{k})$ does not converge.

L1L2

2.1

The tail quotients $q_{j} : = a_{j + 1} / b_{j + 1} = s_{j + 1} (j + 1)^{- 1}$ satisfy $∣ q_{j} ∣ = (j + 1)^{- 1}$ ; given a real $ε > 0$ and a natural $m \geq 1$ with $1 / m < ε$ , every $j \geq m$ has $∣ q_{j} - 0 ∣ = (j + 1)^{- 1} < ε$ , so $q_{j} \to 0$ .

step 1.1step 1.2L4L6

2.2

$b_{k + 1} - b_{k} = 1$ for every $k$ , so $d_{k} = a_{k + 1} - a_{k} = s_{k + 1} - s_{k} = - s_{k} - s_{k} = - 2 s_{k}$ , which is $- 2$ when $k$ is even and $2$ when $k$ is odd.

step 1.1step 1.2L1L3

3.1

$(d_{k})$ does not converge: were $d_{k} \to M$ , then $s_{k} = (- 1 / 2) d_{k}$ would converge to $- M / 2$ by the scalar-multiple rule, contradicting step 1.2.

step 1.2step 2.2L5

4.1

So $(b_{k})$ is strictly increasing and unbounded, the quotients $a_{k} / b_{k}$ converge to $0$ over the indices $k \geq 1$ , and the difference quotients do not converge: the converse of Stolz-Cesaro is false.

step 2.1step 3.1L7∎

Remarks

The implication is genuinely one-way, and the reason is averaging. The conclusion of Stolz-Cesaro, $\infty / \infty$ form: if $b_{k}$ is strictly increasing and unbounded and $(a_{k + 1} - a_{k}) / (b_{k + 1} - b_{k}) \to L$ then $a_{k} / b_{k} \to L$ is obtained by summing the difference quotients against the weights $b_{k + 1} - b_{k}$ and dividing by $b_{n}$ , which is a weighted average. An average can converge while what is averaged oscillates, and here it does: the quotients $a_{k} / b_{k}$ are damped by the growing denominator, and no information about $d_{k}$ survives.
The same phenomenon, in the summability language. With $b_{k} = k$ the weights are equal, so this is the Cesaro situation of FALSE: if the Cesaro means of a sequence converge then the sequence converges and The Cesaro means of $(- 1)^{k}$ converge to $0$ although the sequence diverges in another costume; the divergent object is the same alternating sequence.
The index range is not a technicality. $b_{0} = 0$ , so $a_{0} / b_{0}$ does not denote anything, and the quotient sequence exists only from $k = 1$ . That is why Stolz-Cesaro, $\infty / \infty$ form: if $b_{k}$ is strictly increasing and unbounded and $(a_{k + 1} - a_{k}) / (b_{k + 1} - b_{k}) \to L$ then $a_{k} / b_{k} \to L$ is stated for the tail, and why the convergence asserted above is asserted over $k \geq 1$ .

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Sources

Stolz-Cesàro theorem (Wikipedia) (standard reference, not scraped)
Limit of a sequence (Wikipedia) (standard reference, not scraped)

ak=(−1)kak​=(−1)k, bk=kbk​=k have ak/bk→0ak​/bk​→0 while the difference quotient oscillates, so Stolz-Cesaro has no converse