Geometric Series

Definition

A geometric series is a type of infinite series where each term is a constant multiple of the preceding term. It is a sequence of numbers where the ratio between consecutive terms is constant.

5 Must Know Facts For Your Next Test

1. The formula for the sum of a finite geometric series is $S_n = a(1 - r^n) / (1 - r)$, where $a$ is the first term and $r$ is the common ratio.
2. A geometric series converges if the common ratio $|r| < 1$, and diverges if $|r| \geq 1$.
3. The sum of an infinite geometric series with $|r| < 1$ is given by $S = a / (1 - r)$.
4. Geometric series have many applications in probability, finance, and other areas of mathematics.
5. The geometric distribution, a discrete probability distribution, is closely related to the concept of a geometric series.

Review Questions

• Explain the relationship between a geometric series and the geometric distribution.
  • The geometric distribution, a discrete probability distribution, is closely related to the concept of a geometric series. In the geometric distribution, the random variable represents the number of trials until the first success, where the probability of success in each trial is constant. This is analogous to the terms in a geometric series, where each term is a constant multiple of the preceding term. The formula for the sum of a finite geometric series is directly applicable in calculating probabilities in the geometric distribution.
• Describe the conditions for convergence and divergence of a geometric series.
  • The convergence or divergence of a geometric series is determined by the value of the common ratio, $r$. If $|r| < 1$, the series converges to a finite sum given by $S = a / (1 - r)$, where $a$ is the first term. If $|r| \geq 1$, the series diverges and does not have a finite sum. The common ratio represents the constant ratio between consecutive terms in the series, and its absolute value being less than 1 is the necessary and sufficient condition for the series to converge.
• Analyze the practical applications of geometric series in various fields.
  • Geometric series have numerous practical applications in various fields, including probability, finance, and engineering. In probability, the geometric distribution, which models the number of trials until the first success, is closely related to the concept of a geometric series. In finance, geometric series are used to calculate present values, annuities, and other financial instruments. In engineering, geometric series are used to model exponential growth or decay processes, such as in the analysis of electronic circuits and the propagation of electromagnetic waves. The versatility of geometric series makes them a fundamental tool in many areas of mathematics and its applications.

Related terms

Infinite Series: An infinite series is the sum of an infinite number of terms, where each term is a function of the index of the series.

Common Ratio: The common ratio in a geometric series is the constant ratio between consecutive terms.

Convergence: Convergence in a geometric series refers to the series approaching a finite sum as the number of terms increases.