---
tag: 0Q7K4AXE53
kind: example
name: The sequence r^n tends to zero for |r| < 1
msc: [40A05]
instance_of: [0W4CN9E2W5]
uses: [0MNM54WB99, 0XKWSSY8P2, 0XMF00T4XQ]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_progression
    locator: Properties
---

## 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|0XMF00T4XQ]] gives $|r^n| \to 0$, so $r^n \to 0$.

## Remarks

The [[geometric sequence|0MNM54WB99]] $a r^n$ 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|0TZK91NQE3]].

## Explanations

### L1 human

Multiply a number by $1/2$ again and again: $1, 1/2, 1/4, 1/8, \dots$.
The result gets closer and closer to $0$. The same happens for each factor between $-1$ and $1$.