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A series with ratio limit exactly $1$ that Raabe decides

Example

Take $a_k := 1/\iota(k+1)^{2}$ for $k \in \mathbb{N}$ , so that $\sum a_k$ is $\sum_{k \ge 1} 1/k^{2}$ (Series, partial sums, convergence and the sum, divergence, and the tail series). Then:

its ratios $q_k = a_{k+1}/a_k = \dfrac{(k+1)^{2}}{(k+2)^{2}}$ converge to $1$ , so neither half of the ratio test applies (Ratio test: $\limsup |a_{k+1}/a_k| < 1$ gives absolute convergence and hence convergence, and $\liminf |a_{k+1}/a_k| > 1$ gives divergence);
its Raabe expression is exactly $R_k \;=\; (k+1)\left(\frac{a_k}{a_{k+1}} - 1\right) \;=\; \frac{(k+2)^{2} - (k+1)^{2}}{k+1} \;=\; \frac{2k+3}{k+1} \;=\; 2 + \frac{1}{k+1} ,$ which exceeds $2$ at every index, so $\liminf_k R_k \ge 2 > 1$ and Raabe's test gives convergence (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).

This is the smallest honest illustration that Raabe's test decides series the ratio test cannot. The verdict agrees with For rational $p > 0$ , $\sum 1/k^p$ converges iff $p > 1$ at $p = 2$ , as it must.

Facts & Assumptions

Given: The sequence $a_k := 1/\iota(k+1)^{2}$ , $k \in \mathbb{N}$ ; its ratios $q_k = a_{k+1}/a_k$ ; and its Raabe expression $R_k = (k+1)(a_k/a_{k+1} - 1)$ (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, Integer powers $a^m$ , Canonical naturals are positive and strictly increasing).

[L1]

The canonical naturals are positive, so every $a_k$ is positive; reciprocation on the positives is order reversing; and $x^{2} = x\cdot x$ (Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order, Integer powers $a^m$ , Monotonicity of $x \mapsto x^n$ and of $n \mapsto a^n$ ).

[L2]

For every real $\varepsilon > 0$ there is a natural $n \ge 1$ with $1/n < \varepsilon$ , so $1/\iota(k+1) \to 0$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ , Limits and Cauchy sequences of reals).

[L3]

Algebra of limits: sums, products and quotients of convergent sequences converge, the quotient requiring a nonzero limit and nonzero denominators (Algebra of limits: sums, scalar multiples, products and quotients).

[L4]

The ratio test: its convergence half needs $\limsup_k q_k < 1$ and its divergence half needs $\liminf_k q_k > 1$ (Ratio test: $\limsup |a_{k+1}/a_k| < 1$ gives absolute convergence and hence convergence, and $\liminf |a_{k+1}/a_k| > 1$ gives divergence).

[L5]

Raabe's test: $\liminf_k R_k > 1$ gives convergence (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).

[L6]

Limit superior and inferior in $\overline{\mathbb{R}}$ , their existence for every sequence, and the descriptions $\limsup_k x_k = \inf\{s_n\}$ , $\liminf_k x_k = \sup\{i_n\}$ of the tail bounds (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, 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).

[L7]

$\sum_{k \ge 1} 1/k^{p}$ converges if and only if $p > 1$ (For rational $p > 0$ , $\sum 1/k^p$ converges iff $p > 1$ ).

Verification

technique · direct

1.1

Every $a_k = 1/\iota(k+1)^{2}$ is positive, so the ratios and the Raabe expression are defined.

givenL1

2.1

The ratios are $q_k = \dfrac{1/\iota(k+2)^{2}}{1/\iota(k+1)^{2}} = \dfrac{\iota(k+1)^{2}}{\iota(k+2)^{2}} = \left(1 - \dfrac{1}{\iota(k+2)}\right)^{2}$ .

step 1.1L1algebra

2.2

The Raabe expression is $R_k = (k+1)\left(\dfrac{\iota(k+2)^{2}}{\iota(k+1)^{2}} - 1\right) = \dfrac{\iota(k+2)^{2} - \iota(k+1)^{2}}{\iota(k+1)} = \dfrac{2\iota(k)+3}{\iota(k)+1} = 2 + \dfrac{1}{\iota(k+1)}$ .

step 1.1L1algebra

3.1

Since $1/\iota(k+2) \to 0$ , the product rule gives $q_k \to (1-0)^{2} = 1$ .

step 2.1L2L3

3.2

From step 2.2, $R_k > 2$ for every $k \in \mathbb{N}$ , the added term $1/\iota(k+1)$ being positive.

step 2.2L1

4.1

The convergence half of the ratio test does not apply: if $\limsup_k q_k < 1$ , then with $t$ real and $\limsup_k q_k < t < 1$ some tail supremum would be below $t$ , putting $q_k \le t < 1$ for all large $k$ and contradicting $q_k \to 1$ .

step 3.1L4L6

4.2

The divergence half does not apply either: if $\liminf_k q_k > 1$ , some tail infimum would exceed $1$ , putting $q_k \ge c > 1$ for all large $k$ and again contradicting $q_k \to 1$ .

step 3.1L4L6

4.3

On the other hand $2$ is a lower bound of $\{R_k : k \ge 0\}$ , so the tail infimum $i_0 \ge 2$ and $\liminf_k R_k \ge 2 > 1$ .

step 3.2L6

5.1

Raabe's test therefore gives convergence of $\sum a_k$ , that is of $\sum_{k \ge 1} 1/k^{2}$ , in agreement with the case $p = 2$ of the $p$ -series theorem.

step 4.3step 1.1L5L7∎

Remarks

The Raabe expression here is exact, not asymptotic. Step 2.2 computes $R_k = 2 + 1/(k+1)$ on the nose, so no limit is needed to apply the test: a single inequality $R_k > 2$ at every index already forces $\liminf_k R_k \ge 2$ . That is why this witness is the cleanest available one.
Why the ratio test must fail here. The ratios of any $p$ -series tend to $1$ whatever $p$ is, so a criterion reading only $\limsup$ and $\liminf$ of the ratios cannot separate the convergent $p$ -series from the divergent ones. Raabe reads the rate at which the ratios approach $1$ , which is exactly the missing information, and that rate is $2/k$ up to smaller terms when $p = 2$ .

Depends on

Used by

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Sources

Raabe's test (Wikipedia) (standard reference, not scraped)
Ratio test (Wikipedia) (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)
John K. Hunter, An Introduction to Real Analysis (standard reference, not scraped)

A series with ratio limit exactly 111 that Raabe decides