Geometric series convergence calculator.
a) Arithmetic
b) Geometric
c) Trigonometry
d) Calculus
The Correct Answer and Explanation is:
The correct answer is b) Geometric.
A geometric series is a series of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r). For example, in the geometric series 1 + 1/2 + 1/4 + 1/8 + …, each term is multiplied by 1/2. To determine whether a geometric series converges or diverges, we rely on specific conditions tied to the common ratio.
Convergence Criteria for Geometric Series
A geometric series will converge (approach a finite sum) if and only if the absolute value of the common ratio, |r|, is less than 1. This is expressed mathematically as:
- Convergence condition: If |r| < 1, the series converges.
- Divergence condition: If |r| ≥ 1, the series diverges.
If a geometric series converges, its sum can be calculated using the following formula:
[ S = \frac{a}{1 – r} ]
Where:
- ( S ) is the sum of the series.
- ( a ) is the first term of the series.
- ( r ) is the common ratio.
Example
Consider the geometric series: 3 + 1.5 + 0.75 + …
Here, the first term ( a = 3 ) and the common ratio ( r = 0.5 ). Since |r| = 0.5 < 1, the series converges. The sum of the infinite series is calculated as:
[
S = \frac{3}{1 – 0.5} = \frac{3}{0.5} = 6
]
Why Geometric Series Convergence Is Important
Geometric series are used in many applications, such as in economics for calculating interest, in physics for analyzing wave behavior, and in computer science for algorithms and data structures. Understanding whether a series converges helps in making precise predictions and calculations.