Geometric series

Definition

A geometric series is a series of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The behavior of a geometric series depends heavily on the value of the common ratio and is closely tied to concepts of convergence and divergence, as it determines whether the series approaches a finite limit or grows indefinitely.

5 Must Know Facts For Your Next Test

1. A geometric series can be expressed in the form $$S = a + ar + ar^2 + ar^3 + ...$$ where 'a' is the first term and 'r' is the common ratio.
2. If the absolute value of the common ratio $$|r| < 1$$, the geometric series converges to the sum $$S = \frac{a}{1 - r}$$.
3. If $$|r| \geq 1$$, the geometric series diverges, meaning that its sum does not approach any finite number.
4. The convergence of a geometric series allows for practical applications in finance, physics, and computer science, where it is used to calculate present and future values.
5. The concept of geometric series extends to infinite series, emphasizing how understanding convergence and divergence is crucial for determining whether an infinite sum has a meaningful value.

Review Questions

• How does the common ratio influence the convergence or divergence of a geometric series?
  • The common ratio determines how each term in a geometric series relates to its predecessor. If the absolute value of the common ratio is less than one ($$|r| < 1$$), then the terms decrease in magnitude and approach zero, leading to convergence. Conversely, if the common ratio is greater than or equal to one ($$|r| \geq 1$$), the terms do not decrease and either grow larger or oscillate, resulting in divergence.
• What formula can be used to find the sum of a converging geometric series, and under what conditions is this formula applicable?
  • The sum of a converging geometric 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 is applicable only when the absolute value of the common ratio is less than one ($$|r| < 1$$). If this condition is not met, the series will diverge, and this formula cannot be used.
• Evaluate how understanding geometric series can impact real-world applications like finance and physics.
  • Understanding geometric series has significant implications in various real-world applications. In finance, for instance, they are used to calculate present and future values of investments based on interest rates, which often form geometric progressions. Similarly, in physics, concepts such as decay processes or wave functions can exhibit behaviors modeled by geometric series. Recognizing whether these series converge or diverge helps predict outcomes effectively in these fields.

Related terms

Common Ratio: The fixed number that each term in a geometric series is multiplied by to get the next term.

Convergence: The property of a series where the sum approaches a finite limit as more terms are added.

Divergence: The property of a series where the sum does not approach a finite limit, often growing indefinitely or oscillating.