Math 2.0

statement proposition

Term test

Statement

Let ∑an\sum a_n be a convergent series of real numbers. Then an→0a_n \to 0 as n→∞n \to \infty. Equivalently, if (an)(a_n) does not converge to 00, then the series ∑an\sum a_n diverges.

Remarks

For the proof, let ss be the sum. Then an=sn−sn−1→s−s=0a_n = s_n - s_{n-1} \to s - s = 0. The converse is false. The terms of the harmonic series tend to 00, and the series diverges. For example, ∑(−1)n\sum (-1)^n and ∑n/(n+1)\sum n/(n+1) diverge by this test.

Explanations

L1 human

If you keep adding terms that do not get small, the total cannot settle down. So small terms are necessary for convergence, but they are not enough.

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)

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To read this node, read these 10 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 sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementPartial sum, definition, no evidence levelPartial sumdefinitionLimit of a sum of sequences, statement, no evidence levelLimit of a sum of seq…statementConvergent series, definition, no evidence levelConvergent seriesdefinitionTerm test, statement, no evidence levelTerm 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. Limit of a sequence definition
  4. Series definition
  5. Convergent sequence definition
  6. Epsilon-N definition of the limit of a sequence statement
  7. Partial sum definition
  8. Limit of a sum of sequences statement
  9. Convergent series definition
  10. Term test statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Nth-term_test, Usage
  2. Web https://en.wikipedia.org/wiki/Nth-term_test, Proofs

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