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. This series can be expressed in a closed form, which is particularly useful for calculating the sum when the number of terms is large. The convergence or divergence of the series depends on the value of the common ratio, making it an important concept in various mathematical applications.

5 Must Know Facts For Your Next Test

1. The formula for the sum of the first n terms of a geometric series can be expressed as $$S_n = a \frac{1 - r^n}{1 - r}$$ where 'a' is the first term and 'r' is the common ratio.
2. If the absolute value of the common ratio |r| < 1, an infinite geometric series converges to $$S = \frac{a}{1 - r}$$.
3. If |r| >= 1, an infinite geometric series diverges and does not have a finite sum.
4. Geometric series are often used in real-world applications such as calculating compound interest and modeling exponential growth or decay.
5. Mathematical induction can be used to prove formulas related to geometric series, especially for establishing properties and identities involving their sums.

Review Questions

• How can you use mathematical induction to prove a formula for the sum of a geometric series?
  • To use mathematical induction to prove a formula for the sum of a geometric series, start by establishing the base case for n=1, showing that it holds true. Then, assume that it holds for some arbitrary n=k, leading to the conclusion that it must also hold for n=k+1. By substituting into the formula and simplifying, you can demonstrate that if it works for k, it must work for k+1. This method validates that the formula applies to all natural numbers.
• What conditions must be met for an infinite geometric series to converge, and how does this relate to its common ratio?
  • For an infinite geometric series to converge, the absolute value of its common ratio must be less than 1 (|r| < 1). If this condition is satisfied, the sum approaches a specific limit given by $$S = \frac{a}{1 - r}$$. If |r| is equal to or greater than 1, the series diverges, meaning it does not approach a finite value. Understanding this concept helps clarify when we can expect meaningful results from summing an infinite number of terms.
• Discuss how geometric series can model real-world scenarios such as compound interest and population growth.
  • Geometric series are essential in modeling real-world phenomena like compound interest and population growth because these situations often involve repeated multiplication over time. For compound interest, each period's interest accumulates on both the principal and previously earned interest, forming a geometric sequence. Similarly, population growth can follow a pattern where each generation's size grows by a fixed percentage compared to its predecessor. Both cases can be analyzed using geometric series formulas to project future values accurately and understand long-term trends.

Related terms

geometric sequence: A 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.

common ratio: The common ratio is the factor by which each term in a geometric sequence is multiplied to obtain the next term.

infinite series: An infinite series is the sum of an infinite number of terms from a sequence, which may converge to a specific value or diverge without approaching any limit.