Problem 44 - Algebra and Trigonometry

Question

Find the sum of each infinite geometric series. $$ \sum_{i=1}^{\infty} 12(-0.7)^{j-1} $$

Step-by-Step Solution

Answer

The sum of the infinite geometric series ∑_(i=1)^(∞) \( 12(-0.7)^{j-1} \) is approximately 7.05.

Step 1: Identify the first term and common ratio

From the provided infinite geometric series, it can be deduced that the first term (a) is 12 and the common ratio (r) is -0.7.

Step 2: Apply the infinite geometric series formula

The formula for the sum S of an infinite geometric series is given by \( S = \frac{a}{1 - r} \). Substituting the values \( a = 12 \) and \( r = -0.7 \) into this formula yields \( S = \frac{12}{1 - (-0.7)} \) which becomes \( S = \frac{12}{1.7} \)

Step 3: Evaluate the expression

The expression \( S = \frac{12}{1.7} \) simplifies to approximately \( S ≈ 7.05 \). Hence, the sum of the infinite geometric series is 7.05.

Key Concepts

Geometric Series Formula

The geometric series formula is a critical tool used to find the sum of a geometric series. A geometric series is a sequence of terms in which each term after the first is obtained by multiplying the previous one by a fixed, non-zero number called the "common ratio." When the series is infinite, we have an infinite geometric series.
The formula to determine the sum of an infinite geometric series is:

• \[ S = \frac{a}{1 - r} \]

where \( S \) is the sum of the series, \( a \) is the first term, and \( r \) is the common ratio.
This formula applies only when the absolute value of the common ratio \(| r | < 1|\). If \(| r | \) is greater than or equal to one, the series does not have a finite sum, as the terms do not get smaller and smaller. Understanding this formula is essential for solving problems involving infinite geometric series.

Common Ratio

The concept of the common ratio is pivotal in understanding geometric series. The common ratio, often represented by \( r \), is the factor by which we multiply each term of the series to get the next term.
For example, if the series starts with a term \( a \), the subsequent terms will be \( ar, ar^2, ar^3, \) and so on.
A positive common ratio makes all terms have the same sign, while a negative ratio leads to alternating signs.

• If \( |r| < 1 \), the terms of the series get progressively smaller and approach zero.
• If \( |r| \geq 1 \), the terms become larger or oscillate, making the series diverge.

Understanding whether the common ratio is less than one or not helps us determine if we can find the sum of an infinite series using the geometric series formula.

Sum of Infinite Series

The sum of an infinite series is only calculable under certain conditions. When dealing with an infinite geometric series, we can only find a finite sum if the common ratio's absolute value is less than one.
Here's why: When the series progresses, each term becomes smaller if \( |r| < 1 \), converging towards zero.
This leads to terms gradually contributing less to the sum, allowing the series to approach a finite value.

• For example, with a first term \( a \) and a common ratio \( r \), the sum \( S \) is calculated using the formula: \( S = \frac{a}{1 - r} \)
• In our initial example with \( a = 12 \) and \( r = -0.7 \), you find that \( S = \frac{12}{1.7} \approx 7.05 \).

These terms diminish in size quicker as they alternate between positive and negative, ultimately ensuring the sum converges to approximately 7.05.

First Term

The first term of a geometric series, denoted by \( a \), captures the starting point of the series. It is crucial as it sets the initial value for the rest of the terms.
In the geometric sequence, each subsequent term is a transformation of this first term, thanks to the common ratio.
In the specific series given in the exercise, the first term is \( a = 12 \), serving as the base for calculating the sum of the series.

• The first term is essential in employing the geometric series formula: \( S = \frac{a}{1 - r} \).
• Knowing the first term allows you to easily plug it into the formula to compute the series sum.

Thus, identifying this first term correctly is the premier step in solving problems related to infinite geometric series.