$\sum_{j \ge 0} (-1)^{j}/(j+1)$ converges conditionally, with sum strictly between $1/2$ and $1$

Let $(\varepsilon_j)$ be the alternating sequence (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 $s \circ o \equiv -1$), written $\varepsilon_j = (-1)^j$, and put $b_j := 1/\iota(j+1)$, with $\iota(j+1)$ the canonical natural, positive for every $j$ (Canonical naturals are positive and strictly increasing). The alternating harmonic series is

$\sum_{j \ge 0} \frac{(-1)^{j}}{j+1} \;=\; \sum_j \varepsilon_j b_j .$

It converges conditionally (Absolutely convergent and conditionally convergent series, and the general starting index): it converges, by the alternating series test, while its series of absolute values is the harmonic series $\sum_{k\ge1} 1/k$, which diverges (For rational $p > 0$, $\sum 1/k^p$ converges iff $p > 1$). Writing $S$ for its sum,

$\tfrac{1}{2} \;<\; \tfrac{7}{12} \;\le\; S \;\le\; \tfrac{5}{6} \;<\; 1 .$

The value of $S$ is not asserted. The classical evaluation is a logarithm and is not available at this point in the reading order; what is proved here is that $S$ exists and where it lies. See Selected sums and products on this page that are proved to exist without being evaluated, and what their evaluation waits for.

This is the series that gives the whole page its content: it is the standard witness for FALSE: every convergent series converges absolutely and, through The Riemann series theorem: a conditionally convergent real series has, for every $c \in \mathbb{R}$, a rearrangement with sum $c$, and rearrangements diverging to $+\infty$, to $-\infty$, and oscillating with any prescribed $\liminf \le \limsup$ in $\overline{\mathbb{R}}$, the source of every rearrangement example below.

• The bracketing is exactly the error bound of the test, used twice. Any pair of an even-index and an odd-index partial sum brackets $S$, and the further out the pair is taken the tighter the bracket becomes; $t_4$ and $t_3$ are simply the first pair whose values separate $S$ strictly from $1/2$ and from $1$. Taking $t_2 = 1/2$ and $t_1 = 1$ would give only the non-strict bounds.

• Conditional convergence is a statement about cancellation. The terms have absolute value $1/(j+1)$ and their sum without signs is infinite; the series converges only because consecutive terms nearly cancel. Everything that follows on this page, that the terms may be reordered to sum to anything at all, is a consequence of exactly that.

• What the bracket does not say. It gives no rate and no closed form. Better numerical bounds come from later pairs $t_{e_j}, t_{o_j}$ and cost only arithmetic; the closed form costs the logarithm.