Math 2.0

Sequences, series, summability Proof

Proof of Abel's test from Dirichlet's test

Builds on Convergent seriesMonotone sequenceBounded sequenceMonotone convergence theorem for sequencesDecreasing sequence bounded below convergesDirichlet's testConvergent sequences are boundedLimit of a sum of sequencesLimit of a product of sequencesPartial sum

Proof

We reduce the test to Dirichlet's test.

  1. Let ∑an\sum a_n be a convergent series with partial sums AN=∑n≤NanA_N = \sum_{n \le N} a_n and sum AA. Let (bn)(b_n) be monotone and bounded.
  2. If (bn)(b_n) is increasing, it is bounded above, so it converges by the monotone convergence theorem.
  3. If (bn)(b_n) is decreasing, it is bounded below, so it converges by the decreasing form of that theorem.
  4. Let bb be the limit of (bn)(b_n) and put dn=bn−bd_n = b_n - b. Then (dn)(d_n) is monotone, of the same type as (bn)(b_n), and dn→0d_n \to 0.
  5. The sequence (AN)(A_N) converges, so it is bounded. That is, there is KK with ∣AN∣≤K\abs{A_N} \le K for all NN.
  6. By steps 4 and 5, Dirichlet's test applies to the sequences (dn)(d_n) and (an)(a_n), after a shift of the index by one. So ∑dnan\sum d_n a_n converges, with partial sums DN=∑n≤Nan(bn−b)D_N = \sum_{n \le N} a_n (b_n - b).
  7. For each NN, ∑n≤Nanbn=DN+bAN\sum_{n \le N} a_n b_n = D_N + b A_N, since anbn=an(bn−b)+bana_n b_n = a_n (b_n - b) + b a_n.
  8. The sequence (bAN)(b A_N) converges to bAb A, by the product rule for limits with the constant sequence bb.
  9. By the sum rule for limits, the partial sums DN+bAND_N + b A_N converge. So ∑anbn\sum a_n b_n converges.

Idea

A bounded monotone sequence has a limit bb. Subtracting bb gives a monotone sequence that tends to 00, which is exactly the hypothesis of Dirichlet's test. The leftover part b∑anb \sum a_n converges because ∑an\sum a_n does.

Reading path

To understand this page from the ground up, read these 92 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof that convergent sequences are Cauchy, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of Dirichlet's test by summation by parts, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof that a decreasing sequence bounded below converges, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences.

  1. Construction of the reals from Cauchy sequences concept
  2. Definition by recursion concept
  3. Element definition
  4. Integers object
  5. Mathematical induction concept
  6. Natural numbers object
  7. Ordered field definition
  8. Peano axioms concept
  9. Proposition concept
  10. Rational numbers object
  11. Real numbers object
  12. Sequence definition
  13. Set bounded above definition
  14. Set definition
  15. Ring of integers statement
  16. Successor function definition
  17. Implication definition
  18. Predicate definition
  19. The reals form a field statement
  20. Real sequence definition
  21. Subsequence definition
  22. Infimum definition
  23. Supremum definition
  24. Ordered pair definition
  25. Addition of natural numbers definition
  26. Universal quantifier definition
  27. The successor function is injective statement
  28. Zero is not a successor statement
  29. The rationals form a field statement
  30. Bounded sequence definition
  31. Cauchy sequence definition
  32. Limit of a sequence definition
  33. Monotone sequence definition
  34. Series definition
  35. Completeness axiom statement
  36. Cartesian product definition
  37. Subset definition
  38. Proof of the commutativity of addition by induction proof
  39. Order on the natural numbers definition
  40. The rationals form an ordered field statement
  41. Convergent sequence definition
  42. Partial sum definition
  43. Binary relation definition
  44. Order on the reals definition
  45. Commutativity of addition statement
  46. Proof of strong induction from ordinary induction proof
  47. Convergent series definition
  48. Antisymmetric relation definition
  49. Reflexive relation definition
  50. Transitive relation definition
  51. Strong induction statement
  52. Preorder definition
  53. Proof of the well-ordering principle from strong induction proof
  54. Partial order definition
  55. Well-ordering principle statement
  56. Total order definition
  57. Proof that every sequence has a monotone subsequence, by peaks proof
  58. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  59. Every sequence has a monotone subsequence statement
  60. The reals form an ordered field statement
  61. Absolute value definition
  62. Proof of the approximation property of the supremum proof
  63. Proof of the triangle inequality for real numbers proof
  64. Proof that the absolute value is multiplicative proof
  65. Proof of the epsilon-N definition of the limit of a sequence proof
  66. Approximation property of the supremum statement
  67. Triangle inequality statement
  68. Absolute value is multiplicative statement
  69. Epsilon-N definition of the limit of a sequence statement
  70. Proof of the limit of a sum of sequences proof
  71. Proof that convergent sequences are bounded proof
  72. Proof that convergent sequences are Cauchy proof
  73. Proof that Cauchy sequences are bounded proof
  74. Proof of the monotone convergence theorem for sequences proof
  75. Limit of a sum of sequences statement
  76. Convergent sequences are bounded statement
  77. Convergent sequences are Cauchy statement
  78. Cauchy sequences are bounded statement
  79. Monotone convergence theorem for sequences statement
  80. Proof of the limit of a product of sequences proof
  81. Proof that a decreasing sequence bounded below converges proof
  82. Decreasing sequence bounded below converges statement
  83. Limit of a product of sequences statement
  84. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  85. Bolzano–Weierstrass theorem statement
  86. Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
  87. Cauchy criterion for sequences statement
  88. Proof of the Cauchy criterion for series proof
  89. Cauchy criterion for series statement
  90. Proof of Dirichlet's test by summation by parts proof
  91. Dirichlet's test statement
  92. Proof of Abel's test from Dirichlet's test proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Abel%27s_Test, Proof
About this page
TagPMRSJG1XY4
URIm2://PMRSJG1XY4
Kindproof
WritingW0 stub
MSC40A05
ProvesAbel's test
UsesConvergent series, Monotone sequence, Bounded sequence, Monotone convergence theorem for sequences, Decreasing sequence bounded below converges, Dirichlet's test, Convergent sequences are bounded, Limit of a sum of sequences, Limit of a product of sequences, Partial sum
Changed4 November 2026

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