The period-three pattern $1, 1, -2$ has partial sums in $\{0,1,2\}$, so $\sum a_k/(k+1)$ converges by Dirichlet's test although the alternating series test does not apply

Let $(g_k)$ be the sequence of naturals with values in $\{0,1,2\}$ defined by the recursion $g_0 = 0$ and $g_{k+1} = g_k + 1$ for $g_k \in \{0,1\}$, $g_{k+1} = 0$ for $g_k = 2$ (The recursion theorem), and put

$a_k := \begin{cases} 1 & \text{if } g_k \in \{0,1\}, \\ -2 & \text{if } g_k = 2, \end{cases} \qquad b_k := \frac{1}{\iota(k+1)} .$

So $(a_k)$ is the repeating pattern $1, 1, -2, 1, 1, -2, \dots$ Its partial sums $A_n = \sum_{k<n} a_k$ take only the values $0, 1, 2$, hence are bounded (Lower bound, bounded below, bounded set), while $(b_k)$ is nonincreasing with $b_k \to 0$. By Dirichlet's test (Dirichlet's test: if the partial sums of $\sum a_k$ are bounded and $(b_k)$ is nonincreasing with $b_k \to 0$, then $\sum a_k b_k$ converges) the series

$\sum_k \frac{a_k}{k+1}$

converges. It converges conditionally (Absolutely convergent and conditionally convergent series, and the general starting index), since $|a_k| \ge 1$ for every $k$ and $\sum 1/\iota(k+1)$ is the harmonic series.

The alternating series test does not reach this example. The alternating series test: if $(b_k)$ is nonincreasing with $b_k \to 0$ then $\sum_{k} (-1)^{k} b_k$ converges, the sum lies between any two consecutive partial sums, and the error after $n$ terms is at most $b_n$ is a statement about $\sum \varepsilon_k b_k$ for the alternating sequence $(\varepsilon_k)$, whose values strictly alternate in sign; here $a_0 = a_1 = 1$, so $(a_k)$ is not that sequence, nor any constant multiple of it, and no reading of the test applies. This is the item on the page showing that Dirichlet's test is strictly stronger than the Leibniz criterion, and an alternating witness would not show it.

• Dirichlet's test needs only boundedness of the partial sums. Here they cycle through $0, 1, 2$ and never converge, so $\sum a_k$ itself diverges; the test nevertheless applies, and that is precisely what distinguishes it from Abel's test: if $\sum a_k$ converges and $(b_k)$ is monotone and bounded then $\sum a_k b_k$ converges, which requires $\sum a_k$ to converge.

• Why the pattern is $1, 1, -2$ and not $1, -1$. The blocks must have mean zero for the partial sums to stay bounded, and they must not alternate, or the example would be covered by the alternating series test. The smallest integer pattern with both properties has period three.

• The parity object is not used, and could not be. The even and odd index maps and the alternating sequence: strictly increasing $e, o$ with $\mathbb{N}$ their disjoint union, and the unique $(s_k)$ with $s_0 = 1$, $s_{\sigma(k)} = -s_k$, which satisfies $|s_k| = 1$, $s \circ e \equiv 1$ and $s \circ o \equiv -1$ supplies the period-two structure of this library; a period-three pattern needs its own recursion, which is what the state set $\{0,1,2\}$ provides. No claim is made that the two constructions are instances of a common one.