Math 2.0

statement theorem

Ratio test

Statement

Let (an)(a_n) be a real sequence with an≠0a_n \ne 0 for all nn. Assume that ∣an+1∣/∣an∣→L|a_{n+1}| / |a_n| \to L as n→∞n \to \infty, for a real number LL with L<1L < 1. Then the series ∑an\sum a_n converges absolutely.

Remarks

Choose qq with L<q<1L < q < 1. For n≥Nn \ge N we get ∣an+1∣≤q∣an∣|a_{n+1}| \le q |a_n|, so ∣an∣≤∣aN∣qn−N|a_n| \le |a_N| q^{n-N}. The comparison test with a geometric series gives the result. For L>1L > 1 the series diverges, as the divergence part of the ratio test states. For L=1L = 1 the test gives no answer. The series ∑1/n\sum 1/n diverges and ∑1/n2\sum 1/n^2 converges, and both have L=1L = 1. For example, ∑xn/n!\sum x^n / n! converges for each real xx, since the ratio is ∣x∣/(n+1)→0|x|/(n+1) \to 0.

Explanations

L1 human

If each term is eventually less than a fixed fraction q<1q < 1 of the term before, the terms shrink at least as fast as qnq^n. So the series behaves at least as well as a convergent geometric series.

What this node needs for W2
  • edge about out: the concepts that the statement is about (0 of 1)
  • edge proves in: at least one proof node (0 of 1)
  • edge instance_of or illustrates in: one example that shows the statement at work (0 of 1)
  • explanation L2 (0 of 1)

The writing levels say how complete the text of a node is. They do not measure its truth.

What this needs

To read this node, read these 20 nodes in this order. The foundations come first, and the list ends with this node.

Sequence, definition, no evidence levelSequencedefinitionReal sequence, definition, no evidence levelReal sequencedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementPartial sum, definition, no evidence levelPartial sumdefinitionThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…exampleConvergent series, definition, no evidence levelConvergent seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleRatio test, statement, no evidence levelRatio teststatement
Each node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • no evidence level
  1. Sequence definition
  2. Real sequence definition
  3. Bounded sequence definition
  4. Geometric sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Series definition
  8. Sum of a finite geometric series statement
  9. Monotone convergence theorem for sequences statement
  10. Convergent sequence definition
  11. Squeeze theorem statement
  12. Partial sum definition
  13. The sequence r^n tends to zero for |r| < 1 example
  14. Convergent series definition
  15. A series of nonnegative terms converges iff its partial sums are bounded statement
  16. Absolute convergence definition
  17. Sum of a series definition
  18. Comparison test statement
  19. Geometric series example
  20. Ratio test statement

What depends on this

77 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Ratio_test, The test

What links here

Used by