Infinite geometric sequence

Definition

An infinite geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The sequence continues indefinitely without terminating.

5 Must Know Facts For Your Next Test

1. The formula for the $n$-th term of an infinite geometric sequence is $a_n = a_1 \cdot r^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio.
2. If $|r| < 1$, the sum of an infinite geometric series converges to $\frac{a_1}{1 - r}$.
3. If $|r| \geq 1$, the sum of an infinite geometric series does not converge.
4. The common ratio can be any real number except zero.
5. To determine if a series converges, evaluate whether the absolute value of the common ratio is less than one.

Review Questions

• What is the formula to find any term in an infinite geometric sequence?
• When does the sum of an infinite geometric series converge?
• How do you determine whether a given infinite geometric series converges or diverges?

Related terms

Geometric Sequence: A sequence in which each term after the first is found by multiplying the previous term by a constant called the common ratio.

Common Ratio: The fixed, non-zero number that each term in a geometric sequence is multiplied by to get the next term.

Convergent Series: A series whose terms approach a specific value as more terms are added.