Math 2.0

example

Geometric series

Example

Let aa and rr be real numbers with ∣r∣<1|r| < 1. The geometric series ∑n=0∞arn\sum_{n=0}^{\infty} a r^n converges, and ∑n=0∞arn=a1−r.\sum_{n=0}^{\infty} a r^n = \frac{a}{1 - r}. The sum of a finite geometric series gives the partial sums sN=a(1−rN+1)/(1−r)s_N = a (1 - r^{N+1})/(1 - r). Since rN+1→0r^{N+1} \to 0, the partial sums tend to a/(1−r)a/(1 - r).

Remarks

For a≠0a \ne 0 and ∣r∣≥1|r| \ge 1, the series diverges by the term test, because ∣arn∣≥∣a∣|a r^n| \ge |a|. For a=1a = 1 and r=1/2r = 1/2, it gives 1+1/2+1/4+⋯=21 + 1/2 + 1/4 + \cdots = 2. The decimal 0.333…0.333\ldots is the geometric series with a=3/10a = 3/10 and r=1/10r = 1/10, and its sum is 1/31/3. The terms form a geometric sequence.

Explanations

L1 human

Walk half of the way to a wall, then half of the rest, and so on. The distances 1/2+1/4+1/8+⋯1/2 + 1/4 + 1/8 + \cdots add up to the whole distance 11.

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To read this node, read these 13 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 sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…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 seriesdefinitionSum of a series, definition, no evidence levelSum of a seriesdefinitionGeometric series, example, no evidence levelGeometric seriesexample
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. Geometric sequence definition
  4. Limit of a sequence definition
  5. Series definition
  6. Sum of a finite geometric series statement
  7. Convergent sequence definition
  8. Squeeze theorem statement
  9. Partial sum definition
  10. The sequence r^n tends to zero for |r| < 1 example
  11. Convergent series definition
  12. Sum of a series definition
  13. Geometric series example

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Sources

  1. Web https://en.wikipedia.org/wiki/Geometric_series, Infinite geometric series and convergence

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