Math 2.0

Sequences, series, summability Proof

Proof of the term test

Builds on Convergent seriesPartial sumSum of a seriesLimit of a sequenceTriangle inequality

Proof

  1. Let sN=∑n=0Nans_N = \sum_{n=0}^{N} a_n be the partial sums. Since the series is a convergent series, sN→Ss_N \to S for a real number SS, the sum of the series.
  2. For each n≥1n \ge 1, the partial sums satisfy an=sn−sn−1a_n = s_n - s_{n-1}.
  3. Let ε>0\varepsilon > 0. By the definition of the limit, there is NN with ∣sm−S∣<ε/2\abs{s_m - S} < \varepsilon / 2 for all m≥Nm \ge N.
  4. Let n≥N+1n \ge N + 1. Then both n≥Nn \ge N and n−1≥Nn - 1 \ge N.
  5. By step 2 and the triangle inequality, ∣an∣=∣(sn−S)+(S−sn−1)∣≤∣sn−S∣+∣sn−1−S∣\abs{a_n} = \abs{(s_n - S) + (S - s_{n-1})} \le \abs{s_n - S} + \abs{s_{n-1} - S}.
  6. By steps 3 and 4, the right side of step 5 is less than ε/2+ε/2=ε\varepsilon/2 + \varepsilon/2 = \varepsilon.
  7. So ∣an−0∣<ε\abs{a_n - 0} < \varepsilon for all n≥N+1n \ge N + 1. Since ε>0\varepsilon > 0 was arbitrary, an→0a_n \to 0.
  8. The second form is the contrapositive of the first. If (an)(a_n) does not converge to 00, the series cannot converge, by steps 1 to 7. So it diverges.

Idea

Each term is the difference of two consecutive partial sums. When the partial sums settle down to a limit, consecutive ones get close to each other, so their difference tends to 00.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first. The path takes the proofs Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the triangle inequality for real numbers.

Show the path as a graph
Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRational numbers, object, no evidence levelRational numbersobjectReal numbers, object, no evidence levelReal numbersobjectSequence, definition, no evidence levelSequencedefinitionSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionThe reals form a field, statement, no evidence levelThe reals form a fieldstatementReal sequence, definition, no evidence levelReal sequencedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionTotal order, definition, no evidence levelTotal orderdefinitionProof that the reals form an ordered field, with the Cauchy sequence construction, proof, no evidence levelProof that the reals …proofThe reals form an ordered field, statement, no evidence levelThe reals form an ord…statementAbsolute value, definition, no evidence levelAbsolute valuedefinitionProof of the triangle inequality for real numbers, proof, no evidence levelProof of the triangle…proofTriangle inequality, statement, no evidence levelTriangle inequalitystatementProof of the term test, proof, no evidence levelProof of the term testproof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Rational numbers object
  7. Real numbers object
  8. Sequence definition
  9. Set definition
  10. Ring of integers statement
  11. Implication definition
  12. Predicate definition
  13. The reals form a field statement
  14. Real sequence definition
  15. Ordered pair definition
  16. The rationals form a field statement
  17. Universal quantifier definition
  18. Limit of a sequence definition
  19. Series definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Convergent sequence definition
  24. Partial sum definition
  25. Binary relation definition
  26. Order on the reals definition
  27. Convergent series definition
  28. Antisymmetric relation definition
  29. Reflexive relation definition
  30. Transitive relation definition
  31. Sum of a series definition
  32. Preorder definition
  33. Partial order definition
  34. Total order definition
  35. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  36. The reals form an ordered field statement
  37. Absolute value definition
  38. Proof of the triangle inequality for real numbers proof
  39. Triangle inequality statement
  40. Proof of the term test proof

Where this leads

443 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Term_test
About this page
TagSZ5WPYWAHQ
URIm2://SZ5WPYWAHQ
Kindproof
WritingW0 stub
MSC40A05
ProvesTerm test
UsesConvergent series, Partial sum, Sum of a series, Limit of a sequence, Triangle inequality
Changed4 November 2026

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