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 general form of an infinite geometric sequence is $a, ar, ar^2, ar^3, \ldots$, where $a$ is the first term and $r$ is the common ratio.
2. If the absolute value of the common ratio $|r| < 1$, the infinite geometric series converges to a sum.
3. The sum of an infinite geometric series with $|r| < 1$ can be calculated using the formula $S = \frac{a}{1 - r}$.
4. If $|r| \geq 1$, the infinite geometric series does not converge and thus has no sum.
5. Infinite geometric sequences are used in various fields including finance for calculating perpetuities and in computer science for analyzing algorithms.

Review Questions

• What is the formula to find the sum of an infinite geometric series when $|r| < 1$?
• Explain why an infinite geometric series diverges if $|r| \geq 1$.
• Given a sequence with terms: 3, 1.5, 0.75,... identify if it forms an infinite geometric sequence and find its common ratio.

Related terms

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

Convergence: A property of a series where its terms approach a specific value as more terms are added.

Geometric Series: $\sum_{n=0}^{\infty} ar^n$, represents the sum of an infinite number of terms of a geometric sequence.