Math 2.0

Sequences, series, summability Proof

Proof of the divergence part of the ratio test

Builds on Limit of a sequenceTerm test

Proof

  1. Apply the definition of the limit with ε=L−1>0\varepsilon = L - 1 > 0. There is NN with ∣an+1∣/∣an∣>L−ε=1\abs{a_{n+1}}/\abs{a_n} > L - \varepsilon = 1 for all n≥Nn \ge N.
  2. So ∣an+1∣>∣an∣\abs{a_{n+1}} > \abs{a_n} for n≥Nn \ge N. By induction, ∣an∣≥∣aN∣\abs{a_n} \ge \abs{a_N} for all n≥Nn \ge N.
  3. By hypothesis aN≠0a_N \ne 0, so δ=∣aN∣>0\delta = \abs{a_N} > 0.
  4. If an→0a_n \to 0, then ∣an∣<δ\abs{a_n} < \delta for all large nn. This contradicts step 2.
  5. So (an)(a_n) does not converge to 00. By the term test, the series ∑an\sum a_n diverges.

Idea

When the ratio tends to a number above 11, the absolute values of the terms eventually grow. Growing nonzero terms cannot tend to 00, so the series fails the term test.

Reading path

To understand this page from the ground up, read these 46 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the term test.

  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. Proof of the epsilon-N definition of the limit of a sequence proof
  40. Triangle inequality statement
  41. Epsilon-N definition of the limit of a sequence statement
  42. Proof of the limit of a sum of sequences proof
  43. Proof of the term test proof
  44. Limit of a sum of sequences statement
  45. Term test statement
  46. Proof of the divergence part of the ratio test proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ratio_test, Proof
About this page
TagKAX5R5F099
URIm2://KAX5R5F099
Kindproof
WritingW0 stub
MSC40A05
ProvesDivergence part of the ratio test
UsesLimit of a sequence, Term test
Changed4 November 2026

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