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

Builds on Geometric sequence Limit of a sequence Squeeze theorem

Example

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

Remarks

The geometric sequence $a r^n$ then tends to $0$ for each real $a$ . For $r = 1$ the sequence is constant. For $r = -1$ it is $1, -1, 1, \dots$ , which diverges. For $|r| > 1$ the sequence $|r^n|$ tends to infinity.

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Sequence definition
Real sequence definition
Geometric sequence definition
Limit of a sequence definition
Squeeze theorem statement
The sequence r^n tends to zero for |r| < 1 example

Where this leads

109 pages build on this one.

Tag	`0Q7K4AXE53`
URI	`m2://0Q7K4AXE53`
Kind	example
Writing	W0 stub
MSC	40A05
Instance of	Convergent sequence
Uses	Geometric sequence, Limit of a sequence, Squeeze theorem

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

Example

Remarks

Reading path

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