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 $b_n$

Let $({\epsilon }_k)$ be the alternating sequence of 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 $\mathrm{\mid }{s}_k\mathrm{\mid }=1$, $s\circ e\equiv 1$ and $s\circ o\equiv -1$, that is the unique sequence of reals with ${\epsilon }_0=1$ and ${\epsilon }_{k+1}=-{\epsilon }_k$, which is what is usually written ${\epsilon }_k=(-1{)}^k$; let $e$ and $o$ be its even and odd index maps, so that ${\epsilon }_{e_j}=1$, ${\epsilon }_{o_j}=-1$, and every natural number is $e_j$ for exactly one $j$ or $o_j$ for exactly one $j$.

Let $(b_k)$ be a sequence of reals that is nonincreasing (Nondecreasing, increasing, nonincreasing, decreasing, monotone, and eventually monotone sequences) and converges to $0$ (Limits and Cauchy sequences of reals); then $b_k\ge 0$ for every $k$. Write $t_n:={\sum }_{k<n}{\epsilon }_k{b}_k$ for the partial sums (Series, partial sums, convergence and the sum, divergence, and the tail series). Then:

1. the series $\sum {\epsilon }_k{b}_k$ converges; write $L$ for its sum;
2. $t_{e_j}\le L\le t_{o_j}$ for every $j\in \mathbb{N}$, and for every $n\in \mathbb{N}$ the sum $L$ lies between the two consecutive partial sums $t_n$ and $t_{n+1}$;
3. $\mathrm{\mid }L-t_n\mathrm{\mid }\le b_n$ for every $n\in \mathbb{N}$.

Claim 3 is the error bound: the partial sum $t_n$, which uses the $n$ terms ${\epsilon }_0{b}_0,\dots ,{\epsilon }_{n-1}{b}_{n-1}$, differs from the sum by at most the first term omitted.

Only claim 1 is a corollary of 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. Claims 2 and 3 are not: they come from the interlacing of the even-index and odd-index partial sums, and that argument is carried out below rather than smuggled into the Dirichlet estimate, which produces no bracketing at all.

• The two hypotheses are not interchangeable with "$b_k\to 0$" alone. A null sequence that is not monotone can make $\sum {\epsilon }_k{b}_k$ diverge, and the bracketing of step 3.1 is exactly where monotonicity enters; the error bound is false without it. The test as stated is the classical Leibniz criterion.

• Why the index maps rather than "$2j$" and "$2j+1$". The even and odd index maps come from 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 $\mathrm{\mid }{s}_k\mathrm{\mid }=1$, $s\circ e\equiv 1$ and $s\circ o\equiv -1$ together with the parity object itself, and step 1.3 is the only arithmetic needed about them. Rebuilding $(-1{)}^k$ by a fresh recursion inside this proof, and then proving afresh that the even indices and the odd indices partition $\mathbb{N}$, is precisely what that lemma exists to prevent.

• What the test does not give. It produces the sum $L$ as a limit and bounds the error, and it identifies $L$ with no closed expression. For the alternating harmonic series the value is not available at this point in the reading order; see Selected sums and products on this page that are proved to exist without being evaluated, and what their evaluation waits for.