Math 2.0

definition

Conditional convergence

Definition

Let (an)(a_n) be a real sequence. The series ∑an\sum a_n converges conditionally if ∑an\sum a_n converges and ∑∣an∣\sum |a_n| diverges.

Remarks

So a convergent series converges either absolutely or conditionally, and not both. The alternating harmonic series converges conditionally. By the Riemann rearrangement theorem, the terms of a conditionally convergent series can be reordered to give any real sum.

Explanations

L1 human

A conditionally convergent series converges only because its positive and negative terms cancel. The positive terms alone add up to +∞+\infty, and the negative terms alone add up to −∞-\infty.

What this node needs for W2
  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (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 9 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 sequencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionAbsolute convergence, definition, no evidence levelAbsolute convergencedefinitionConditional convergence, definition, no evidence levelConditional convergen…definition
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. Limit of a sequence definition
  4. Series definition
  5. Convergent sequence definition
  6. Partial sum definition
  7. Convergent series definition
  8. Absolute convergence definition
  9. Conditional convergence definition

What depends on this

1 node uses this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Conditional_convergence, Definition

What links here

Used by