Math 2.0

statement theorem

Comparison test

Statement

Let (an)(a_n) and (bn)(b_n) be real sequences with 0≤an≤bn0 \le a_n \le b_n for all nn. If ∑bn\sum b_n converges, then ∑an\sum a_n converges, and ∑an≤∑bn\sum a_n \le \sum b_n. If ∑an\sum a_n diverges, then ∑bn\sum b_n diverges.

Remarks

The partial sums of ∑an\sum a_n are at most those of ∑bn\sum b_n. The result follows from a series of nonnegative terms converges iff its partial sums are bounded. It is enough that 0≤an≤bn0 \le a_n \le b_n for all nn larger than some NN. For example, 0≤1/(n2+1)≤1/n20 \le 1/(n^2 + 1) \le 1/n^2, so ∑1/(n2+1)\sum 1/(n^2 + 1) converges. Mathlib states it for a summable upper bound, in Summable.of_nonneg_of_le.

Explanations

L1 human

If a series of nonnegative terms is smaller term by term than a series with a finite sum, it has a finite sum too. If a smaller series already has an infinite sum, the larger one has an infinite sum as well.

What this node needs for W2
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  • edge proves in: at least one proof node (0 of 1)
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  • explanation L2 (0 of 1)

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To read this node, read these 12 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 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 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.
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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

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Sources

  1. Web https://en.wikipedia.org/wiki/Direct_comparison_test, For series

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