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What is an Infinite Geometric Series?
An infinite geometric series is a series where the terms follow a geometric pattern and the number of terms is infinite. In other words, it's a sequence of numbers where each term is found by multiplying the previous term by a constant factor, called the common ratio, and this pattern continues indefinitely.
For example, the series 1 + 1/2 + 1/4 + 1/8 + ... is an infinite geometric series where the common ratio is 1/2. Each term is half of the previous term.
The Formula for the Sum of an Infinite Geometric Series
The sum of an infinite geometric series can be calculated using the following formula:
S = a / (1 - r)
Where:
- S is the sum of the infinite series
- a is the first term of the series
- r is the common ratio
However, this formula only works if the absolute value of the common ratio (|r|) is less than 1 (i.e., -1 < r < 1). If |r| is greater than or equal to 1, the series diverges, meaning the sum will go to infinity.
Understanding Convergence and Divergence
An infinite geometric series is said to be convergent if its sum approaches a finite value. This occurs when the absolute value of the common ratio is less than 1. In this case, the series has a defined limit.
On the other hand, an infinite geometric series is said to be divergent if its sum grows infinitely large. This happens when the absolute value of the common ratio is greater than or equal to 1. The series doesn't have a defined limit.
Example: Finding the Sum of an Infinite Geometric Series
Let's find the sum of the infinite geometric series: 2 + 1 + 1/2 + 1/4 + ...
- The first term (a) is 2.
- The common ratio (r) is 1/2 (each term is half of the previous term).
Using the formula, we get:
S = a / (1 - r) = 2 / (1 - 1/2) = 2 / (1/2) = 4
Therefore, the sum of the infinite geometric series 2 + 1 + 1/2 + 1/4 + ... is 4.
Applications of Infinite Geometric Series
Infinite geometric series have applications in various fields, including:
- Physics: Modeling the decay of radioactive substances
- Finance: Calculating the present value of an infinite stream of payments
- Calculus: Evaluating limits of certain functions
Key Takeaways
- An infinite geometric series is a series where each term is found by multiplying the previous term by a constant factor (common ratio).
- The sum of an infinite geometric series can be calculated using the formula S = a / (1 - r) if |r| < 1.
- Series converge if |r| < 1 and diverge if |r| ≥ 1.
Understanding the concept of infinite geometric series and its formula is crucial for solving various mathematical problems and understanding real-world phenomena.