Math 2.0

example

The sequence r^n tends to zero for |r| < 1

Example

Let rr be a real number with ∣r∣<1|r| < 1. Then rn→0r^n \to 0 as n→∞n \to \infty. For r=0r = 0 this is clear. For 0<∣r∣<10 < |r| < 1, write 1/∣r∣=1+h1/|r| = 1 + h with h>0h > 0. By Bernoulli's inequality, (1+h)n≥1+nh>nh(1 + h)^n \ge 1 + n h > n h for each n≥1n \ge 1. Thus 0≤∣rn∣<1/(nh)0 \le |r^n| < 1/(n h), and the squeeze theorem gives ∣rn∣→0|r^n| \to 0, so rn→0r^n \to 0.

Remarks

The geometric sequence arna r^n then tends to 00 for each real aa. For r=1r = 1 the sequence is constant. For r=−1r = -1 it is 1,−1,1,…1, -1, 1, \dots, which diverges. For ∣r∣>1|r| > 1 the sequence ∣rn∣|r^n| tends to infinity.

Explanations

L1 human

Multiply a number by 1/21/2 again and again: 1,1/2,1/4,1/8,…1, 1/2, 1/4, 1/8, \dots. The result gets closer and closer to 00. The same happens for each factor between −1-1 and 11.

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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 sequencedefinitionSqueeze theorem, statement, no evidence levelSqueeze theoremstatementThe sequence r^n tends to zero for |r| < 1, example, no evidence levelThe sequence r^n tend…example
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  1. Sequence definition
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  1. Web https://en.wikipedia.org/wiki/Geometric_progression, Properties

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