Selected sums and products on this page that are proved to exist without being evaluated, and what their evaluation waits for

A convergence test proves that a limit exists; it does not produce the limit. On this page that gap is systematic, and this remark records the principal places where a familiar value or formula is deferred and what would close it. Every scope statement below is relative to the reading order: the material named is developed elsewhere in this library, later than this page, and nothing here says it is absent from the library.

The alternating harmonic series. 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 $b_n$ proves that $\sum_{j \ge 0} (-1)^j/(j+1)$ converges, and its error bound pins the sum between consecutive partial sums; the companion examples page uses that to prove the sum lies strictly between $1/2$ and $1$. No closed expression for the sum is given, and none can be given here: the classical value is a logarithm, and the logarithm is introduced later in the reading order. So the sum is named, bracketed, and left unevaluated.

The two-positive-one-negative rearrangement. The same is true one level up. The companion examples page proves that taking two positive terms for each negative one produces a convergent rearrangement whose sum is $3/2$ times the sum of the original series. That statement is exact and complete as it stands, and it is deliberately relative: it compares two sums rather than evaluating either. The familiar form of the same fact multiplies a logarithm by $3/2$, and it becomes available at the same later point.

The refined criterion for infinite products. For $p_k \ge 0$ the product $\prod (1 + p_k)$ converges iff $\sum p_k$ converges, with $1 + \sum_{k<n} p_k \le \prod_{k<n}(1+p_k) \le 1/\bigl(1 - \sum_{k<n} p_k\bigr)$ when $\sum_{k<n} p_k < 1$; for $0 \le p_k < 1$ the product $\prod (1 - p_k)$ converges iff $\sum p_k$ converges and its partial products tend to $0$ otherwise; and $\sum |p_k|$ convergent implies $\prod (1+p_k)$ convergent settles $\prod(1+p_k)$ completely for $p_k \ge 0$, settles $\prod(1-p_k)$ for $0 \le p_k < 1$, and proves that $\sum |p_k|$ convergent forces $\prod(1+p_k)$ convergent. It does not settle the remaining case: a signed sequence $(p_k)$ with $\sum p_k$ convergent but $\sum |p_k|$ divergent. The classical criterion there is that $\prod(1+p_k)$ converges exactly when $\sum p_k^{2}$ converges. A standard proof expands $\log(1+x)$; that route belongs with the logarithm, later in the reading order. The gap is not hypothetical: the companion examples page exhibits a signed sequence with $\sum p_k$ convergent whose partial products tend to $0$.

Rearrangement beyond $\mathbb{R}$. 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 $\overline{\mathbb{R}}$ and For a series of real numbers, unconditional convergence and absolute convergence are the same property together answer the rearrangement question for real series completely. The corresponding question for series of vectors is raised, and left open at this point in the reading order, in The same question in $\mathbb{R}^d$: what the set of rearrangement sums looks like, and why that answer is not reachable at this point in the reading order, which states no theorem about it.

Two places where existence is constructive but no formula is claimed. Base-$b$ expansions: for an integer $b \ge 2$ every $x \in [0,1)$ is the sum of $\sum_{j \ge 0} d_j / b^{\,j+1}$ for digits $d_j < b$, and the digit sequence is unique among those that are not eventually constantly $b-1$ produces, for every $x \in [0,1)$, its digit sequence in base $b$, by a recursion that depends on $x$; it gives no closed expression for the digits of any particular real, and it claims none. Likewise 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 $\overline{\mathbb{R}}$ produces, for each prescribed target, a bijection of $\mathbb{N}$ defined by a recursion over the terms of the series; no formula for that bijection is given, and the theorem asserts only that one exists. In both cases the construction is fully determined by the data, with no choice made anywhere, which is a stronger statement than mere existence and a weaker one than a formula.

What this list does not claim. It is not a census of every convergence result on the page. In particular, the Dirichlet, alternating-series, and Abel tests and their worked applications establish additional convergence without evaluating a numerical sum; their purpose here is to supply convergence criteria, not to flag a familiar value whose evaluation waits for a later object. Among the structural comparison theorems, Dirichlet's rearrangement theorem: an absolutely convergent series converges unconditionally, and every rearrangement of it has the same sum, Mertens' theorem: if $\sum a_k$ converges absolutely to $A$ and $\sum b_k$ converges to $B$, their Cauchy product converges to $AB$, If $\sum a_k$ and $\sum b_k$ both converge absolutely then their Cauchy product converges absolutely, with sum $AB$, Grouping: if $\sum a_k$ converges and $(n_j)$ is strictly increasing with $n_0 = 0$, the series of blocks $\sum_{k=n_j}^{n_{j+1}-1} a_k$ converges to the same sum and Fubini for double series: if $\sum_i \sum_j |a_{ij}|$ converges then both iterated sums and the sum along every bijection $\mathbb{N} \to \mathbb{N} \times \mathbb{N}$ converge to one and the same value identify sums with one another and evaluate nothing, which is exactly what makes them usable wherever the sums themselves are unknown.