Alphabeta MathHow 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.
If a series converges then its terms tend to
Statement
Let be a sequence of reals. If the series converges (Series, partial sums, convergence and the sum, divergence, and the tail series) then the sequence converges (Limits and Cauchy sequences of reals), with
Equivalently, in the contrapositive form in which the lemma is almost always used: if does not converge to then diverges.
The same statement holds for a series with a general starting index, with the conclusion that converges to : that series is by definition the series of the sequence (Series, partial sums, convergence and the sum, divergence, and the tail series), so it is the statement above applied to .
The converse is false; it is refuted by FALSE: if then converges.
Facts & Assumptions
Given: A sequence of reals whose series converges, with partial sums (Series, partial sums, convergence and the sum, divergence, and the tail series, Finite sums and finite products, by recursion).
The recursion clause of the finite sum: for every (Finite sums and finite products, by recursion, Series, partial sums, convergence and the sum, divergence, and the tail series).
Convergence depends only on the tail: if converges to then so does its -st tail (Convergence depends only on the tail, Sequences of reals: bounded, eventually, frequently, tails, subsequences).
Algebra of limits: if and then (Algebra of limits: sums, scalar multiples, products and quotients).
Proof
The sequence is the -st tail of , and it converges to .
From the recursion clause, for every .
Hence is the difference of two sequences converging to , so it converges to .
That is the claim, and its contrapositive is the assertion that a series whose terms do not tend to diverges.
Remarks
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This is a necessary condition and never a sufficient one. It rules a series out; it never rules one in. The harmonic series has terms tending to and diverges, which is exactly the content of FALSE: if then converges.
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What the proof actually uses. Only that the partial sums converge and that consecutive partial sums differ by a term. No sign hypothesis is placed on , and none is available at this point on the page.
Depends on
- Series, partial sums, convergence and the sum, divergence, and the tail series
- Algebra of limits: sums, scalar multiples, products and quotients
- Limits and Cauchy sequences of reals
- Finite sums and finite products, by recursion
- Convergence depends only on the tail
- Sequences of reals: bounded, eventually, frequently, tails, subsequences
Used by
- Kummer with ζₖ = 1 recovers the ratio test Corollary
- (1-1) + (1-1) + … converges to 0 while ∑ₖ (-1)ᵏ diverges Counterexample
- A nonnegative non-monotone sequence for which ∑ aₖ and ∑ 2ᵏ a_2ᵏ behave differently Counterexample
- Equal radii do not determine convergence on the boundary circle Counterexample
- For every x>2 the uncontinued Mercator series at u=x-1 diverges Counterexample
- The Cauchy product of ∑_k ≥ 0 (-1)ᵏ/√k+1 with itself has |cₙ| ≥ 1 for every n, so it diverges Counterexample
- Trace class operator Definition
- Boundary behavior can agree or differ for Dirichlet-series abscissae Example
- The Cantor set is homeomorphic to {0,1}^ℕ with the product of discrete topologies, the ternary digits being the coordinates Example
- The harmonic series ∑ 1/k diverges, by condensation and by Oresme block grouping Example
- The Hilbert cube [0,1]^ℕ with the product topology is metrizable, by d(x,y) = ∑ₖ |xₖ - yₖ| / 2^ k+1 Example
- The lacunary power series with factorial exponents has radius one and diverges at 1 Example
- FALSE: convergence of a complex power series at one point other than its centre forces convergence everywhere False statement
- FALSE: if aₖ → 0 then ∑ aₖ converges False statement
- FALSE: if some grouping of a series converges then the series itself converges False statement
- FALSE: the Cauchy product of two convergent series converges False statement
- A power series, its formal derivative, and its zero-constant-term formal antiderivative have the same radius of convergence Lemma
- Base-b expansions: for an integer b ≥ 2 every x ∈ [0,1) is the sum of ∑_j ≥ 0 dⱼ / b^ j+1 for digits dⱼ < b, and the digit sequence is unique among those that are not eventually constantly b-1 Theorem
- For |r| < 1, ∑_k ≥ 0 rᵏ = 1/(1-r), and for |r| ≥ 1 the series diverges Theorem
- Kummer: for positive terms aₖ and weights ζₖ > 0, liminf(ζₖ aₖ/aₖ₊₁ - ζₖ₊₁) > 0 gives convergence, and if ∑ 1/ζₖ diverges while that expression is eventually ≤ 0 the series diverges Theorem
- Ratio test: limsup |aₖ₊₁/aₖ| < 1 gives absolute convergence and hence convergence, and liminf |aₖ₊₁/aₖ| > 1 gives divergence Theorem
- Root test: limsup |aₖ|^1/k < 1 gives absolute convergence and hence convergence, > 1 gives divergence, and = 1 decides nothing Theorem
- The p-series for a real exponent p converges exactly when p is greater than one Theorem
- The Riemann series theorem: a conditionally convergent real series has, for every c ∈ ℝ, a rearrangement with sum c, and rearrangements diverging to +∞, to -∞, and oscillating with any prescribed liminf ≤ limsup in ℝ̄ Theorem
- Under square summability, the signed product of (1+pₙ) converges iff the series of pₙ converges Theorem
Dependency tree · two levels
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Sources
- Term test (Wikipedia) (standard reference, not scraped)
- W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (standard reference, not scraped)
- John K. Hunter, An Introduction to Real Analysis (standard reference, not scraped)