Infinite Geometric Series

Definition

An infinite geometric series is a series where each term is a fixed multiple of the previous term. These series continue on indefinitely, with no end point, and can be used to model various real-world phenomena.

5 Must Know Facts For Your Next Test

1. The sum of an infinite geometric series can be calculated using a formula that involves the first term and the common ratio.
2. Infinite geometric series can either converge to a finite value or diverge to infinity, depending on the value of the common ratio.
3. Convergent infinite geometric series have a common ratio less than 1 in absolute value, while divergent series have a common ratio greater than or equal to 1 in absolute value.
4. Infinite geometric series are often used to model real-world phenomena such as population growth, radioactive decay, and the behavior of electrical circuits.
5. The concept of infinite geometric series is closely linked to the idea of a geometric sequence, as the series is formed by the accumulation of the terms in the sequence.

Review Questions

• Explain the relationship between a geometric sequence and an infinite geometric series.
  • A geometric sequence is a sequence where each term is a fixed multiple of the previous term, creating a pattern of growth or decay. An infinite geometric series is formed by the accumulation of the terms in a geometric sequence, where the series continues on indefinitely. The common ratio between consecutive terms in the sequence determines whether the corresponding infinite series will converge to a finite value or diverge to infinity.
• Describe the conditions under which an infinite geometric series will converge, and how the sum of the series can be calculated.
  • For an infinite geometric series to converge, the common ratio between consecutive terms must be less than 1 in absolute value. When this condition is met, the sum of the infinite series can be calculated using the formula: $S = \frac{a}{1-r}$, where $a$ is the first term and $r$ is the common ratio. This formula allows for the determination of the finite value that the series will approach as the number of terms increases.
• Analyze how the properties of infinite geometric series can be used to model real-world phenomena, and provide examples of such applications.
  • Infinite geometric series can be used to model a variety of real-world phenomena due to their ability to describe patterns of growth or decay. For example, the exponential growth of a population can be modeled using an infinite geometric series, where each term represents the population at a successive time interval. Similarly, the behavior of electrical circuits, such as the charging and discharging of capacitors, can be described using infinite geometric series. Additionally, the concept of infinite geometric series is fundamental to understanding the behavior of radioactive decay, where the series represents the diminishing amount of a radioactive substance over time.

Related terms

Geometric Sequence: A sequence where each term is a fixed multiple of the previous term, creating a pattern of growth or decay.

Common Ratio: The fixed ratio between consecutive terms in a geometric sequence or series.

Convergence: The property of a series where the sum of the terms approaches a finite value as the number of terms increases.