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

Example

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

$\sum_{j \geq 0} \frac{(- 1)^{j}}{j + 1} = \sum_{j} ε_{j} b_{j} .$

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 $\sum_{k \geq 1} 1 / k$ , which diverges (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ ). Writing $S$ for its sum,

$\frac{1}{2} < \frac{7}{12} \leq S \leq \frac{5}{6} < 1.$

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 $c \in R$ , a rearrangement with sum $c$ , and rearrangements diverging to $+ \infty$ , to $- \infty$ , and oscillating with any prescribed $lim inf \leq lim sup$ in $\overline{R}$ , the source of every rearrangement example below.

Facts & Assumptions

Given: The alternating sequence $(ε_{j})$ with index maps $e$ and $o$ , the sequence $b_{j} = 1 / ι (j + 1)$ , and the partial sums $t_{n} = \sum_{j < n} ε_{j} b_{j}$ (Series, partial sums, convergence and the sum, divergence, and the tail series).

[L1]

The alternating sequence: $ε_{0} = 1$ , $ε_{j + 1} = - ε_{j}$ , $∣ ε_{j} ∣ = 1$ ; $e_{0} = 0$ , $e_{j + 1} = e_{j} + 2$ , $o_{0} = 1$ , $o_{j + 1} = o_{j} + 2$ ; $ε_{e_{j}} = 1$ and $ε_{o_{j}} = - 1$ (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]

The canonical naturals are positive for $n \geq 1$ and strictly increasing; reciprocation reverses the order on the positives; and for every real $ε > 0$ there is $n \geq 1$ with $1 / ι (n) < ε$ (Canonical naturals are positive and strictly increasing, Inverses of positives are positive, and reciprocation reverses order, For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L3]

The alternating series test: for $(b_{j})$ nonincreasing with $b_{j} \to 0$ , $\sum ε_{j} b_{j}$ converges with sum $L$ , and $t_{e_{j}} \leq L \leq t_{o_{j}}$ for every $j$ (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 $b_{n}$ , Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences, Limits and Cauchy sequences of reals).

[L4]

$\sum_{k \geq 1} 1 / k^{p}$ converges if and only if $p > 1$ , with $ι (k)^{1} = ι (k)$ ; and $\sum_{k \geq 1} x_{k}$ is the series of $j \mapsto x_{j + 1}$ (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ , Rational powers $a^{r}$ of a positive base, Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ , Integer powers $a^{m}$ , Series, partial sums, convergence and the sum, divergence, and the tail series).

[L5]

Absolute value: $∣ x y ∣ = ∣ x ∣ ∣ y ∣$ (Basic properties of the absolute value).

[L6]

Partial sums: $t_{0} = 0$ and $t_{n + 1} = t_{n} + ε_{n} b_{n}$ (Series, partial sums, convergence and the sum, divergence, and the tail series, Finite sums and finite products, by recursion, Laws of finite sums and finite products).

[L7]

Absolute and conditional convergence (Absolutely convergent and conditionally convergent series, and the general starting index); limits preserve non-strict inequalities (Limits preserve non-strict inequalities).

Verification

technique · direct

1.1

Every $b_{j} = 1 / ι (j + 1)$ is positive, and $(b_{j})$ is nonincreasing, since $0 < ι (j + 1) < ι (j + 2)$ .

givenL2

1.2

By [L1], $e_{1} = 2$ , $e_{2} = 4$ , $o_{1} = 3$ ; and by [L6] together with $ε_{e_{j}} = 1$ , $ε_{o_{j}} = - 1$ , the first partial sums are $t_{1} = b_{0} = 1$ , $t_{2} = 1 - 1 / 2 = 1 / 2$ , $t_{3} = 1 / 2 + 1 / 3 = 5 / 6$ and $t_{4} = 5 / 6 - 1 / 4 = 7 / 12$ .

L1L6algebra

2.1

$(b_{j})$ converges to $0$ : given a rational $ε > 0$ , take $n \geq 1$ with $1 / ι (n) < ε$ ; for $j \geq n$ one has $ι (j + 1) \geq ι (n) > 0$ , so $b_{j} \leq 1 / ι (n) < ε$ .

step 1.1L2

2.2

For every $j$ , $∣ ε_{j} b_{j} ∣ = ∣ ε_{j} ∣ b_{j} = 1 / ι (j + 1)$ , and $\sum_{j} 1 / ι (j + 1)$ is the $p$ -series $\sum_{k \geq 1} 1 / k$ at $p = 1$ , which diverges.

step 1.1L1L4L5

3.1

By the alternating series test the series converges; write $S$ for its sum, and $t_{e_{j}} \leq S \leq t_{o_{j}}$ holds for every $j$ .

step 1.1step 2.1L3

4.1

Taking $j = 2$ in the lower bound and $j = 1$ in the upper bound of step 3.1 gives $7 / 12 = t_{4} = t_{e_{2}} \leq S \leq t_{o_{1}} = t_{3} = 5 / 6$ .

step 3.1step 1.2

5.1

Since $1 / 2 < 7 / 12$ and $5 / 6 < 1$ , the sum satisfies $1 / 2 < S < 1$ .

step 4.1algebra

6.1

So the series converges while its series of absolute values diverges: it converges conditionally, with sum strictly between $1 / 2$ and $1$ .

step 3.1step 5.1step 2.2L7∎

Remarks

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.

Depends on

Used by

Dependency tree · two levels

66 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Harmonic series (mathematics) (Wikipedia) (standard reference, not scraped)
Alternating series test (Wikipedia) (standard reference, not scraped)
John K. Hunter, An Introduction to Real Analysis (standard reference, not scraped)
N. Donaldson, Math 140A: Series (standard reference, not scraped)

∑j≥0(−1)j/(j+1)∑j≥0​(−1)j/(j+1) converges conditionally, with sum strictly between 1/21/2 and 11