LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Convergent series add and scale termwise

Statement

Let $(a_{k})$ and $(b_{k})$ be sequences of reals whose series converge (Series, partial sums, convergence and the sum, divergence, and the tail series), and let $c \in R$ . Then:

$\sum (a_{k} + b_{k})$ converges, with $\sum_{k = 0}^{\infty} (a_{k} + b_{k}) = \sum_{k = 0}^{\infty} a_{k} + \sum_{k = 0}^{\infty} b_{k}$ ;
$\sum c a_{k}$ converges, with $\sum_{k = 0}^{\infty} c a_{k} = c \sum_{k = 0}^{\infty} a_{k}$ .

Moreover, for $c \neq 0$ and an arbitrary sequence $(a_{k})$ , whose series is not assumed to converge:

$\sum c a_{k}$ converges if and only if $\sum a_{k}$ converges. Equivalently, $\sum a_{k}$ diverges if and only if $\sum c a_{k}$ diverges.

Claim 3 is the form used whenever a comparison is made against a constant multiple of a known series.

Facts & Assumptions

Given: Sequences $(a_{k})$ , $(b_{k})$ of reals and $c \in R$ , with partial sums $s_{n} = \sum_{k < n} a_{k}$ and $t_{n} = \sum_{k < n} b_{k}$ (Series, partial sums, convergence and the sum, divergence, and the tail series, Finite sums and finite products, by recursion).

[L1]

Additivity and scaling of finite sums: $\sum_{k < n} (a_{k} + b_{k}) = \sum_{k < n} a_{k} + \sum_{k < n} b_{k}$ and $\sum_{k < n} c a_{k} = c \sum_{k < n} a_{k}$ (Laws of finite sums and finite products).

[L2]

Algebra of limits: if $x_{n} \to x$ and $y_{n} \to y$ then $x_{n} + y_{n} \to x + y$ and $c x_{n} \to c x$ (Algebra of limits: sums, scalar multiples, products and quotients).

Proof

technique · direct

1.1

The partial sums of $\sum (a_{k} + b_{k})$ are $\sum_{k < n} (a_{k} + b_{k}) = s_{n} + t_{n}$ , and those of $\sum c a_{k}$ are $\sum_{k < n} c a_{k} = c s_{n}$ .

givenL1

1.2

Assume $\sum a_{k}$ and $\sum b_{k}$ converge, say $s_{n} \to S$ and $t_{n} \to T$ .

given

2.1

Then $s_{n} + t_{n} \to S + T$ , so $\sum (a_{k} + b_{k})$ converges with sum $S + T$ , which is claim 1.

step 1.2step 1.1L2

2.2

Likewise $c s_{n} \to c S$ , so $\sum c a_{k}$ converges with sum $c S$ , which is claim 2.

step 1.2step 1.1L2

3.1

For claim 3, let $c \neq 0$ and let $(a_{k})$ be arbitrary. If $\sum a_{k}$ converges then $\sum c a_{k}$ converges by claim 2.

step 2.2

3.2

Conversely, if $\sum c a_{k}$ converges then applying claim 2 to the sequence $(c a_{k})$ and the scalar $1 / c$ , which exists since $c \neq 0$ , shows that $\sum (1 / c) (c a_{k}) = \sum a_{k}$ converges.

step 2.2algebra

4.1

The two implications are claim 3, and its contrapositive form is the statement about divergence.

step 3.1step 3.2∎

Remarks

There is no product rule here, and there is no rule for $\sum a_{k} b_{k}$ . The proof works because a finite sum is additive and homogeneous, and neither property has an analogue for products. Multiplying series is a genuinely harder question, requiring absolute convergence, and it is not treated on this page.
Claim 3 needs $c \neq 0$ and nothing else. In particular it does not need either series to converge, which is what makes it usable in the divergence direction: scaling a divergent series by a nonzero constant leaves it divergent.

Depends on

Used by

Dependency tree · two levels

31 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

Series (mathematics) (Wikipedia) (standard reference, not scraped)
T. Tao, Analysis I, 3rd ed., §7.2 (standard reference, not scraped)
John K. Hunter, An Introduction to Real Analysis (standard reference, not scraped)