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Abel-Dini applied to $\sum 1/k$ : $\sum 1/(k s_k)$ still diverges while $\sum 1/(k s_k^2)$ converges

Example

Take $a_k := 1/\iota(k+1)$ for $k \in \mathbb{N}$ , so that $\sum a_k$ is the harmonic series $\sum_{k \ge 1} 1/k$ , which has positive terms and diverges (The harmonic series $\sum 1/k$ diverges, by condensation and by Oresme block grouping). Its inclusive partial sums are the harmonic numbers

$S_n \;=\; \sum_{k=0}^{n} a_k \;=\; \sum_{k=1}^{n+1} \frac{1}{k} \;=\; H_{n+1} \qquad (n \in \mathbb{N}),$

all of them positive. The Abel-Dini theorem (For a divergent series of positive terms with partial sums $s_k$ , the series $\sum a_k/s_k$ diverges and $\sum a_k/s_k^2$ converges) then says that

$\sum_{n} \frac{a_n}{S_n} \;=\; \sum_{n} \frac{1}{(n+1)\,H_{n+1}} \quad \text{diverges}, \qquad \sum_{n} \frac{a_n}{S_n^{2}} \;=\; \sum_{n} \frac{1}{(n+1)\,H_{n+1}^{2}} \quad \text{converges} .$

Classically these are written $\sum_{k \ge 1} 1/(k H_k)$ and $\sum_{k \ge 1} 1/(k H_k^{2})$ , with $H_k = 1 + 1/2 + \dots + 1/k$ .

What the pair shows. The harmonic series is a familiar slowly divergent explicit series, and dividing its terms by the running total produces something that diverges more slowly still. Dividing by the square of the running total overshoots into convergence. So exponent $1$ gives a divergent member and exponent $2$ a convergent one. The absence of a slowest divergent positive series comes from applying Abel-Dini again to the newly produced divergent series, not from a last-exponent claim about this fixed pair.

Facts & Assumptions

Given: The sequence $a_k := 1/\iota(k+1)$ , $k \in \mathbb{N}$ , and its inclusive partial sums $S_n = \sum_{k=0}^{n} a_k$ (Series, partial sums, convergence and the sum, divergence, and the tail series, Finite sums and finite products, by recursion, Canonical naturals are positive and strictly increasing).

[L1]

The canonical naturals are positive, so each $a_k$ is positive and each $S_n$ is a sum of positive terms (Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order).

[L2]

The harmonic series $\sum_{k \ge 1} 1/k$ diverges (The harmonic series $\sum 1/k$ diverges, by condensation and by Oresme block grouping), and it is by definition the series of $j \mapsto 1/\iota(j+1)$ (Series, partial sums, convergence and the sum, divergence, and the tail series).

[L3]

Abel-Dini: for a sequence of positive terms whose series diverges, with $S_n$ the inclusive partial sums, $\sum_n a_n/S_n$ diverges and $\sum_n a_n/S_n^{2}$ converges (For a divergent series of positive terms with partial sums $s_k$ , the series $\sum a_k/s_k$ diverges and $\sum a_k/s_k^2$ converges, Integer powers $a^m$ ).

Verification

technique · direct

1.1

Every term $a_k = 1/\iota(k+1)$ is positive.

givenL1

1.2

The series $\sum a_k$ is the harmonic series $\sum_{k \ge 1} 1/k$ and therefore diverges.

givenL2

1.3

Its inclusive partial sums are $S_n = \sum_{k=0}^{n} 1/\iota(k+1) = \sum_{k=1}^{n+1} 1/k = H_{n+1}$ , a reindexing of the sum by $k \mapsto k+1$ .

givenL1

2.1

The hypotheses of Abel-Dini are met by $(a_k)$ : positive terms and a divergent series.

step 1.1step 1.2L3

3.1

Therefore $\sum_n \dfrac{a_n}{S_n} = \sum_n \dfrac{1}{\iota(n+1) H_{n+1}}$ diverges.

step 2.1step 1.3L3

4.1

And $\sum_n \dfrac{a_n}{S_n^{2}} = \sum_n \dfrac{1}{\iota(n+1) H_{n+1}^{2}}$ converges.

step 2.1step 1.3L3∎

Remarks

This is the concrete form of the no-slowest-series obstruction. The general statement is that no divergent series of positive terms is eventually dominated by every other; here it is exhibited for the standard candidate. Anyone proposing the harmonic series as a universal comparison series is answered by the first of the two conclusions.
No growth estimate for $H_k$ is used or needed. The classical statement $H_k \approx \log k$ would make both conclusions look like instances of the $p$ -series with a logarithmic correction, but neither the logarithm nor that estimate is available in this library at this point, and the theorem does not require them: it needs only that the running totals are positive, nondecreasing and unbounded.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 84 results over 26 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Harmonic series (mathematics) (Wikipedia) (standard reference, not scraped)
K. Knopp, Theory and Application of Infinite Series, Ch. IX (standard reference, not scraped)
Abel-Dini-Pringsheim theorem (Wikipedia) (standard reference, not scraped)
John K. Hunter, An Introduction to Real Analysis (standard reference, not scraped)

Abel-Dini applied to ∑1/k\sum 1/k∑1/k: ∑1/(ksk)\sum 1/(k s_k)∑1/(ksk​) still diverges while ∑1/(ksk2)\sum 1/(k s_k^2)∑1/(ksk2​) converges