Math 2.0

Sequences, series, summability Convergence tests Theorem

Limit comparison test

Builds on Comparison testLimit of a sequence

Statement

Let (an)(a_n) and (bn)(b_n) be real sequences with an≥0a_n \ge 0 and bn>0b_n > 0 for all nn. Assume that an/bn→ca_n / b_n \to c as n→∞n \to \infty, for a real number cc with 0<c<∞0 < c < \infty. Then ∑an\sum a_n converges if and only if ∑bn\sum b_n converges.

Remarks

For large nn we have c2bn≤an≤2c bn\frac{c}{2} b_n \le a_n \le 2c\, b_n. The comparison test then gives both directions. For example, nn3+1/1n2→1\frac{n}{n^3 + 1} / \frac{1}{n^2} \to 1, so ∑n/(n3+1)\sum n/(n^3 + 1) converges, like ∑1/n2\sum 1/n^2. If c=0c = 0, convergence of ∑bn\sum b_n still implies convergence of ∑an\sum a_n. The converse can fail.

Reading path

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Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementComparison test, statement, no evidence levelComparison teststatementLimit comparison test, statement, no evidence levelLimit comparison teststatement
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.
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  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Limit of a sequence definition
  5. Monotone sequence definition
  6. Series definition
  7. Convergent sequence definition
  8. Monotone convergence theorem for sequences statement
  9. Partial sum definition
  10. Convergent series definition
  11. A series of nonnegative terms converges iff its partial sums are bounded statement
  12. Comparison test statement
  13. Limit comparison test statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Limit_comparison_test, Statement
About this page
Tag0HW7BNHHGF
URIm2://0HW7BNHHGF
Kindstatement, theorem
WritingW1 stated
MSC40A05
StatusThis copy of the site has no build data. Run m2 status 0HW7BNHHGF.
UsesComparison test, Limit of a sequence
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