Geometric series

Definition

A geometric series is a series of terms in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.

5 Must Know Facts For Your Next Test

1. The general form of a geometric series is $a + ar + ar^2 + ar^3 + ...$, where $a$ is the first term and $r$ is the common ratio.
2. A geometric series converges if the absolute value of the common ratio is less than 1 ($|r| < 1$).
3. The sum of an infinite geometric series with $|r| < 1$ can be calculated using $S = \frac{a}{1 - r}$.
4. For a finite geometric series with $n$ terms, the sum can be calculated using $S_n = a \frac{1 - r^n}{1 - r}$ if $r \neq 1$.
5. If the common ratio is greater than or equal to 1, or less than or equal to -1, an infinite geometric series will diverge.

Review Questions

• What is the formula for the sum of an infinite geometric series when it converges?
• How do you determine if a given geometric series converges or diverges?
• Calculate the sum of the first five terms of a geometric series with initial term $3$ and common ratio $0.5$.

Related terms

Arithmetic Series: A sequence of numbers in which each term after the first is found by adding a constant called the common difference.

Common Ratio: The fixed factor between consecutive terms of a geometric sequence.

Convergence: A property indicating that as more terms are added, a series approaches a specific finite value.