Problem 42 - A Graphical Approach to Precalculus with Limits

Question

Use a formula to find the sum of each series. $$\sum_{j=1}^{3}-3\left(\frac{1}{4}\right)^{j}$$

Step-by-Step Solution

Answer

The sum of the series is \(-\frac{63}{64}\).

Step 1: Identify the Type of Series

The series given is \( \sum_{j=1}^{3} -3 \left(\frac{1}{4}\right)^{j} \). This is a geometric series where the first term \( a \) is \( -3 \left(\frac{1}{4}\right) \), the common ratio \( r \) is \( \frac{1}{4} \), and the number of terms \( n \) is 3.

Step 2: Recall the Formula for the Sum of a Geometric Series

The formula for the sum of the first \( n \) terms of a geometric series is \( S_n = a \frac{1-r^n}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio.

Step 3: Compute the First Term

Identify the first term \( a = -3 \left(\frac{1}{4}\right)^1 = -\frac{3}{4} \). This is the value of the series when \( j = 1 \).

Step 4: Use the Formula to Calculate the Sum

Substitute \( a = -\frac{3}{4} \), \( r = \frac{1}{4} \), and \( n = 3 \) into the sum formula: \[ S_3 = -\frac{3}{4} \cdot \frac{1-\left(\frac{1}{4}\right)^3}{1-\frac{1}{4}} \].

Step 5: Simplify the Expression

Calculate \( \left(\frac{1}{4}\right)^3 = \frac{1}{64} \), then substitute it back: \[ S_3 = -\frac{3}{4} \cdot \frac{1-\frac{1}{64}}{\frac{3}{4}} \]. Simplifying, \( S_3 = -\frac{3}{4} \cdot \left(1 - \frac{1}{64}\right) \cdot \frac{4}{3} \).

Step 6: Compute and Simplify Further

Calculate \( 1 - \frac{1}{64} = \frac{63}{64} \). Substitute and simplify: \( S_3 = -\frac{3}{4} \cdot \frac{63}{64} \cdot \frac{4}{3} = -\frac{63}{64} \).

Step 7: Conclude with Final Answer

The sum of the series \( \sum_{j=1}^{3} -3 \left(\frac{1}{4}\right)^{j} \) is \(-\frac{63}{64} \).

Key Concepts

Sum of a series

In mathematics, the sum of a series is essentially the result you get when you add the terms of a sequence together. For geometric series, which involve terms that have a constant ratio, there's a special formula for finding the sum. When you have determined that the series is geometric, like in the given exercise, you can use this formula.
The formula for the sum of the first n terms of a geometric series is:

• \( S_n = a \frac{1-r^n}{1-r} \)

Here, \( S_n \) is the sum of the series, \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
This formula helps in calculating the sum efficiently without having to add each term individually. Just plug in the values for \( a \), \( r \), and \( n \), then simplify the expression to find your sum.
Using this approach provides not just a numerical answer, but also gives insight into the pattern and behavior of the series.

Common ratio

The common ratio in a geometric series is a crucial element that defines how each successive term in the series is determined.
In the given series, the common ratio \( r \) is \( \frac{1}{4} \).
To find the common ratio, take any term in the series and divide it by the previous term.
In our exercise, dividing any term in the form \( -3\left(\frac{1}{4}\right)^j \) by its predecessor results in \( \frac{1}{4} \), confirming that it's consistent throughout the series.

• The common ratio is essential because it helps in determining other properties of the series, such as the sum.
• If the common ratio \( |r| \) is less than 1, the series will approach a sum as you add more terms.
• If \( |r| \) is greater than 1, the series diverges and doesn't settle towards a finite sum.

In this exercise, \( r = \frac{1}{4} \) means it is converging, leading to a finite sum.

First term calculation

The first term of a series is the starting point from which the rest of the terms are generated.
In a geometric series, it is imperative to accurately find this first term, as it directly influences the whole series and the sum you'll calculate.
In the provided exercise, the first term, denoted as \( a \), is calculated by substituting \( j = 1 \) into the series formula:
\( a = -3 \left( \frac{1}{4} \right)^1 = -\frac{3}{4} \).

• To find the first term, simply use the formula given for the series and substitute the smallest index, usually 1.
• This calculation is imperative as \( a \) is the basis for both setting the pattern of the series and in computing the sum using the sum formula.

Calculating the first term allows you to apply the sum of the series formula correctly, resulting in an accurate determination of the series' total value.