Math 2.0

Sequences, series, summability Series: basic notions Example

Geometric series

Builds on Geometric sequenceSum of a seriesSum of a finite geometric seriesThe sequence r^n tends to zero for |r| < 1

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.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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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 page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  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

Where this leads

108 pages build on this one.

Further reading

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