What is an Infinite Geometric Sequence?

A geometric sequence is one where each term is obtained by multiplying the previous term by a fixed value called the common ratio, $r$ . Most of the time in IB Maths, we deal with finite sequences , but sequences can also continue indefinitely.

Infinite Geometric Sequence: A geometric sequence that has no final term , it continues forever. The general term is $u_{n} = u_{1} \cdot r^{n - 1}$ for all positive integers $n$ .

The key question with any infinite sequence is: what happens to the terms as $n \to \infty$ ? Do they grow without bound, or do they settle toward zero?

Example

Consider the sequence with $u_{1} = 1$ and $r = \frac{1}{2}$ :
$1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$
Each term is exactly half the previous one. The terms get closer and closer to zero but never actually reach it. This sequence is said to converge.

Example

Now consider $u_{1} = 2$ and $r = 3$ :
$2, 6, 18, 54, 162, \dots$
The terms grow without bound. This sequence diverges , there is no finite limit.

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What is an Infinite Geometric Sequence?

Infinite Geometric Sequence: A geometric sequence that has no final term , it continues forever. The general term is $u_{n} = u_{1} \cdot r^{n - 1}$ for all positive integers $n$ .

The key question with any infinite sequence is: what happens to the terms as $n \to \infty$ ? Do they grow without bound, or do they settle toward zero?

Example

Now consider $u_{1} = 2$ and $r = 3$ :
$2, 6, 18, 54, 162, \dots$
The terms grow without bound. This sequence diverges , there is no finite limit.

Free preview

AHL 1.11—Sum of infinite geometric sequences

What is an Infinite Geometric Sequence?

8 more sections in this topic

AHL 1.11—Sum of infinite geometric sequences

What is an Infinite Geometric Sequence?

8 more sections in this topic