Geometric series

Definition

A geometric series is the sum of the terms of a geometric sequence, where 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. A geometric series can be finite or infinite.
2. The sum of a finite geometric series can be calculated using the formula $S_n = a \frac{1-r^n}{1-r}$ where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.
3. If $|r| < 1$, an infinite geometric series converges to $\frac{a}{1-r}$.
4. If $|r| \geq 1$, an infinite geometric series diverges.
5. Geometric series are used in various applications including finance for calculating compound interest.

Review Questions

• What is the formula for the sum of a finite geometric series?
• Under what condition does an infinite geometric series converge?
• How do you identify if a given series is geometric?

Related terms

Geometric Sequence:

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

Common Ratio:

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

Convergence:

In mathematics, convergence refers to whether a given infinite series approaches a specific value as more terms are added.