Problem 99 - Precalculus with Limits

Question

In Exercises 93 - 106, find the sum of the infinite geometric series. \( \sum_{n=0}^{\infty}\left(0.4\right)^n \)

Step-by-Step Solution

Answer

The sum of the infinite geometric series \( \sum_{n=0}^{\infty}\left(0.4\right)^n \) is approximately \( S \approx 1.67 \).

Step 1: Identify the ratio and the first term

From the given infinite series \( \sum_{n=0}^{\infty}\left(0.4\right)^n \), it can be identified that the first term (a) is when \( n = 0 \). Since anything raised to the power of 0 equals 1, the first term is 1. The ratio (r) is the term that is raised to the power of n, which is 0.4.

Step 2: Apply the formula for the sum of an infinite geometric series

Now that the values of a and r are identified, they can be substituted into the formula \( S = a/(1 - r) \). So this becomes \( S = 1/(1 - 0.4) \).

Step 3: Calculate the sum

By performing the operations in the denominator first, according to the BIDMAS/BODMAS rule, the equation becomes \( S = 1/0.6 \). Dividing 1 by 0.6 gives the sum of the series as \( S \approx 1.67 \).

Key Concepts

Sum of Series Formula

The sum of an infinite geometric series holds a special place in mathematics as it lets us determine the entirety of an endless sequence. When we talk about infinite geometric series, we refer to the pattern that unfolds like a never-ending stack. Each subsequent number progresses according to a consistent pattern called a ratio. The sum of these numbers, however, does not extend to infinity when the series is convergent.

To find this sum, we use a specific formula that applies to all infinite geometric series where the absolute value of the common ratio is less than 1. This formula is:

• S = \( \frac{a}{1 - r} \)

• S is the sum of the series.
• a is the first term of the series.
• r is the common ratio.

By plugging in these values, you can efficiently compute the series' sum. For instance, if the ratio is 0.4, this formula allows us to calculate the entire sum as a concise number by using simple arithmetic operations. This formula provides a reliable shortcut to manage and make sense of infinite sequences.

Geometric Series First Term

Understanding the first term of a geometric series is foundational in determining the series' behavior. A geometric series consists of terms that grow or shrink by a consistent factor, known as the common ratio. The first term, denoted as \(a\), is where it all begins; it sets the stage for everything that follows.

In our specific example, the series begins with the expression \(\left(0.4\right)^n\). By substituting \(n = 0\), any non-zero number or term raised to the power of zero equals 1. Hence, the first term, \(a\), simplifies to 1.

It's crucial to pinpoint this starting point because it anchors the series, basing all subsequent terms on this initial value. Thus, even if the series lengthens infinitely, identifying the first term equips us with a benchmark for accurately applying the sum of series formula.

Geometric Series Common Ratio

The common ratio is a defining feature of a geometric series. It is the consistent factor by which each term is multiplied to arrive at the next term in the sequence. In essence, the common ratio dictates the pace and direction in which the series evolves.

For the infinite geometric series \( \sum_{n=0}^{\infty}\left(0.4\right)^n \), the common ratio, denoted as \(r\), is identified as 0.4. This can be seen directly from the variable \(\left(0.4\right)^n\), where each term results from multiplying by 0.4 repeatedly.

Choosing the correct common ratio is imperative for implementing the sum of series formula accurately. The series will converge to a finite sum only if the absolute value of \(r\) is less than 1. In this instance, because 0.4 is within this limit, the series cleanly converges, allowing us to compute its total with certainty.