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 eventually, convergence of gives convergence of , and divergence of gives divergence of
Statement
Let and be sequences of reals and suppose there is with
Then:
- if converges then converges (Series, partial sums, convergence and the sum, divergence, and the tail series);
- if diverges then diverges.
The same statement holds verbatim for series with a general starting index , applied to the shifted sequences of Series, partial sums, convergence and the sum, divergence, and the tail series.
The hypothesis is on the terms from some index on, not on all of them: finitely many terms of either sequence may violate it, or be negative, without affecting the conclusion. What may not be dropped is nonnegativity of from that index on.
Facts & Assumptions
Given: Sequences , of reals and with for all ; the partial sums and of the -th tail series (Series, partial sums, convergence and the sum, divergence, and the tail series, Finite sums and finite products, by recursion).
Monotonicity of finite sums: if for all then (Laws of finite sums and finite products).
A series converges if and only if its -th tail series converges (A series converges iff each of its tail series converges, and the sum splits as plus the -th tail).
For a series of nonnegative terms: it converges if and only if the range of its partial sums is bounded above, and in the convergent case every partial sum is at most the sum (A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum, Lower bound, bounded below, bounded set).
Proof
For every the index is at least , so ; in particular both tail series have nonnegative terms.
Assume converges. Then its -th tail series converges.
By monotonicity of finite sums, for every .
That tail series has nonnegative terms, so its partial sums satisfy for every , where is its sum.
Hence for every , so the range of is bounded above by .
The tail series has nonnegative terms and partial sums bounded above, so it converges.
Therefore converges, which is claim 1.
Claim 2 is the contrapositive of claim 1: if diverges then cannot converge.
Remarks
-
Both nonnegativity hypotheses are used, and in different places. is what lets convergence of be read off from boundedness of its partial sums, and is what makes the sum of an upper bound for the partial sums . Drop the sign hypothesis and the theorem is false, not merely unproved; the companion page exhibits a pair with for every , convergent and divergent.
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The comparison is with a series, not with a limit. No quotient appears and no is required to be nonzero, which is what distinguishes this test from the limit comparison test proved next.
Depends on
- A series of nonnegative terms converges iff its partial sums are bounded, and then the sum is their supremum
- A series converges iff each of its tail series converges, and the sum splits as $s_N$ plus the $N$-th tail
- Series, partial sums, convergence and the sum, divergence, and the tail series
- Laws of finite sums and finite products
- Lower bound, bounded below, bounded set
- Finite sums and finite products, by recursion
Used by
- If ∑ aₖ and ∑ bₖ both converge absolutely then their Cauchy product converges absolutely, with sum AB Corollary
- ∏_j ≥ 0 (1 + (-1)ʲ/√j+2) has partial products tending to 0 although ∑_j ≥ 0 (-1)ʲ/√j+2 converges Counterexample
- Two series with aₖ ≤ bₖ for all k, ∑ bₖ convergent and ∑ aₖ divergent, when the terms may be negative Counterexample
- The subspace Γ of directions along which a series converges absolutely, and its orthogonal complement Γ^⊥ Definition
- ∑ 1/k² converges with sum at most 2, by comparison with the telescoping ∑ 1/(k(k-1)) 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 period-three pattern 1, 1, -2 has partial sums in {0,1,2}, so ∑ aₖ/(k+1) converges by Dirichlet's test although the alternating series test does not apply Example
- FALSE: limsup |aₖ₊₁/aₖ| ≥ 1 implies the series diverges False statement
- A power series, its formal derivative, and its zero-constant-term formal antiderivative have the same radius of convergence Lemma
- For 0<x<1, the Abel transform of a series is (1-x)²∑_n≥0(n+1)σₙxⁿ, where σₙ are the Cesaro means of its partial sums Lemma
- If for every ε > 0 some continuous g : X → ℝ satisfies | f(x) - g(x)| < ε for all x, then f is continuous; in particular a uniformly convergent series of continuous real functions has a continuous sum Lemma
- Positive and negative parts: aₖ = aₖ⁺ - aₖ⁻ and |aₖ| = aₖ⁺ + aₖ⁻; a series converges absolutely iff both ∑ aₖ⁺ and ∑ aₖ⁻ converge, and for a conditionally convergent series both diverge to +∞ Lemma
- The Cauchy product of two absolutely convergent complex series converges absolutely to the product of their sums Lemma
- The exponential series converges absolutely for every real argument Lemma
- How the nonnegative tests are ordered by strength, and which of them this page cannot state without the logarithm Remark
- A countable pure-step integrator evaluates a continuous integrand as the absolutely convergent weighted sum of its values at the jumps Theorem
- Abel's limit theorem: if a real series converges to s, then its power series tends to s as x↑1 Theorem
- An absolutely convergent series in ℝⁿ converges, and every rearrangement converges to the same sum Theorem
- Dirichlet's test: if the partial sums of ∑ aₖ are bounded and (bₖ) is nonincreasing with bₖ → 0, then ∑ aₖ bₖ converges Theorem
- Euler's formula: exp(iθ)=cosθ+i sinθ for every real θ Theorem
- For a divergent series of positive terms with partial sums sₖ, the series ∑ aₖ/sₖ diverges and ∑ aₖ/sₖ² converges Theorem
- For aₖ, bₖ > 0 with aₖ/bₖ → L: if L ∈ (0,∞) the two series share their behaviour, while L = 0 and L = ∞ give one implication each Theorem
- Fubini for double series: if ∑ᵢ ∑ⱼ |aᵢⱼ| converges then both iterated sums and the sum along every bijection ℕ → ℕ × ℕ converge to one and the same value Theorem
- Gauss: for positive terms, if aₖ/aₖ₊₁ = 1 + h/k + rₖ with |rₖ| ≤ C k^-1-ε for k ≥ 1, some constant C and some rational ε > 0, the series converges iff h > 1 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 Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series Theorem
- Under square summability, the signed product of (1+pₙ) converges iff the series of pₙ converges Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 61 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Direct comparison test (Wikipedia) (standard reference, not scraped)
- W. Rudin, Principles of Mathematical Analysis, 3rd ed., Ch. 3 (3.25) (standard reference, not scraped)
- John K. Hunter, An Introduction to Real Analysis (standard reference, not scraped)