Math 2.0

example

Decimal expansion as a series

Example

Let (dn)n≥1(d_n)_{n \ge 1} be a sequence of digits, with dn∈{0,1,…,9}d_n \in \{0, 1, \dots, 9\}. The decimal 0.d1d2d3…0.d_1 d_2 d_3 \dots is the sum of the series ∑n=1∞dn10n.\sum_{n=1}^{\infty} \frac{d_n}{10^n}. This series converges, and its sum lies in [0,1][0, 1].

Remarks

The series converges by the comparison test with the geometric series ∑9/10n=1\sum 9/10^n = 1. Two different digit sequences can give the same number. For example, 0.4999…=0.50.4999\ldots = 0.5. Conversely, each real number in [0,1)[0, 1) has a decimal expansion.

Explanations

L1 human

A decimal like 0.3140.314 means 3/10+1/100+4/10003/10 + 1/100 + 4/1000. An infinite decimal is the same idea with infinitely many terms, so it is a series.

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To read this node, read these 19 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 sequencedefinitionGeometric sequence, definition, no evidence levelGeometric sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionSum of a finite geometric series, statement, no evidence levelSum of a finite geome…statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …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 seriesdefinitionA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSum of a series, definition, no evidence levelSum of a seriesdefinitionComparison test, statement, no evidence levelComparison teststatementGeometric series, example, no evidence levelGeometric seriesexampleDecimal expansion as a series, example, no evidence levelDecimal expansion as …example
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. Bounded sequence definition
  4. Geometric sequence definition
  5. Limit of a sequence definition
  6. Monotone sequence definition
  7. Series definition
  8. Sum of a finite geometric series statement
  9. Monotone convergence theorem for sequences statement
  10. Convergent sequence definition
  11. Squeeze theorem statement
  12. Partial sum definition
  13. The sequence r^n tends to zero for |r| < 1 example
  14. Convergent series definition
  15. A series of nonnegative terms converges iff its partial sums are bounded statement
  16. Sum of a series definition
  17. Comparison test statement
  18. Geometric series example
  19. Decimal expansion as a series example

Sources

  1. Web https://en.wikipedia.org/wiki/Decimal_representation, Integer and fractional parts

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