The Sum of an Infinite Geometric Series: A Deep Dive

In the realm of mathematics, geometric series play a crucial role. They are sequences of numbers where each term is found by multiplying the previous term by a constant value, known as the common ratio. But what happens when we consider an infinite geometric series, where the terms continue indefinitely? This is where the concept of the sum of an infinite geometric series comes into play.

Defining the Infinite Geometric Series

An infinite geometric series is a series with an infinite number of terms, where each term is obtained by multiplying the preceding term by a constant common ratio. It can be represented as follows:

a + ar + ar² + ar³ + ...

where:

• a is the first term of the series
• r is the common ratio

Convergence and Divergence

The concept of convergence and divergence is essential when dealing with infinite geometric series. A series is said to be convergent if the sum of its terms approaches a finite value as the number of terms increases infinitely. Conversely, a series is divergent if the sum of its terms does not approach a finite value.

The convergence or divergence of an infinite geometric series depends entirely on the value of the common ratio (r). If the absolute value of the common ratio is less than 1 (|r| < 1), the series converges. If the absolute value of the common ratio is greater than or equal to 1 (|r| ≥ 1), the series diverges.

Calculating the Sum of a Convergent Infinite Geometric Series

For a convergent infinite geometric series, the sum (S) can be calculated using the following formula:

S = a / (1 - r)

where:

• a is the first term
• r is the common ratio

Illustrative Example

Let's consider an example to understand the calculation of the sum of a convergent infinite geometric series. Suppose we have the following infinite geometric series:

1 + 1/2 + 1/4 + 1/8 + ...

In this case, the first term (a) is 1 and the common ratio (r) is 1/2. Since |r| = 1/2 < 1, the series converges. Using the formula, the sum of the series is:

S = 1 / (1 - 1/2) = 1 / (1/2) = 2

Therefore, the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... is 2.

Key Takeaways

• An infinite geometric series is a series with an infinite number of terms, where each term is found by multiplying the previous term by a constant common ratio.
• The convergence or divergence of an infinite geometric series depends on the value of the common ratio. If |r| < 1, the series converges; if |r| ≥ 1, the series diverges.
• The sum of a convergent infinite geometric series can be calculated using the formula S = a / (1 - r).

Understanding the sum of an infinite geometric series is essential in various mathematical applications, from calculus to probability theory. By grasping the concepts of convergence, divergence, and the formula for calculating the sum, you can effectively analyze and solve problems involving infinite geometric series.