Problem 108 - Precalculus with Limits

Question

In Exercises 103-112, use sigma notation to write the sum. \( 1 - \dfrac{1}{2} + \dfrac{1}{4} - \dfrac{1}{8} + \cdots - \dfrac{1}{128} \)

Step-by-Step Solution

Answer

The equivalent sum in sigma notation is \( \sum_{n=1}^{8} \left( -\dfrac{1}{2} \right)^{n-1} \).

Step 1: Identify the common ratio

In geometric series, a key thing to figure out is the common ratio. Here, it can be seen that each term is half of the previous term when considering absolute values. Hence, the absolute value of the common ratio is \( \dfrac{1}{2} \). However, the sign of each term changes for each next term. So to represent both the halving and sign change, the common ratio \( r \) should be \( -\dfrac{1}{2} \).

Step 2: Write the general term using common ratio

Now, for any geometric series with the first term \( a \) and a common ratio \( r \), the \( n \)th term can be written as \( a \cdot r^{n-1} \). Here the first term \( a \) is equal to 1 and the common ratio \( r \) is \( -\dfrac{1}{2} \), so the \( n \)th term can be written as \( 1 \cdot \left( -\dfrac{1}{2} \right)^{n-1} \).

Step 3: Write down the series in sigma notation

The sum as given in the exercise can be now written in standard sigma notation as \( \sum_{n=1}^{8} \left( -\dfrac{1}{2} \right)^{n-1} \). The upper limit of the sum is 8 because \( \dfrac{1}{128} \) is the eighth term when the first term is 1. Hence, the equivalent sum in sigma notation is \( \sum_{n=1}^{8} \left( -\dfrac{1}{2} \right)^{n-1} \).

Key Concepts

Sigma Notation

Sigma notation is a concise and elegant way to represent a series, which is a sum of terms. It uses the Greek letter \( \Sigma \) to indicate summation. This notation is particularly helpful in mathematical expressions because it provides a clear and compact form for representing the addition of a sequence of numbers.

When writing a series in sigma notation, it includes:

• A variable, often \( n \), that represents the index of summation.
• A lower limit, which shows the starting index, like \( n=1 \).
• An upper limit, which indicates where the summation ends, such as 8 in the example.
• An expression for the general term of the series, such as \( \left( -\dfrac{1}{2} \right)^{n-1} \).

For instance, when expressing the sum \( 1 - \dfrac{1}{2} + \dfrac{1}{4} - \dfrac{1}{8} + \cdots - \dfrac{1}{128} \) in sigma notation, we use: \[\sum_{n=1}^{8} \left( -\dfrac{1}{2} \right)^{n-1}\] The notation indicates that we are summing the terms generated from the expression \( \left( -\dfrac{1}{2} \right)^{n-1} \) as \( n \) goes from 1 to 8.

Common Ratio

The common ratio is a crucial part of any geometric sequence. It is the factor by which we multiply one term to get the next term in the series. Recognizing the common ratio is essential for solving geometric series problems, and it forms the basis for understanding how the terms in the sequence progress.

In the given series, each term is half the size and opposite in sign of the previous one. This means that the absolute value of the common ratio is \( \dfrac{1}{2} \), but because the sign alternates with each term, the common ratio \( r \) is indeed \( -\dfrac{1}{2} \).

This common ratio \( r \) helps us to determine any term in the series using the formula:

• The first term \( a = 1 \).
• The common ratio \( r = -\dfrac{1}{2} \).

The \( n \)th term of the sequence can be found using the formula \( a \cdot r^{n-1} \), which in this case becomes \( 1 \cdot \left(-\dfrac{1}{2}\right)^{n-1} \).

Geometric Sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. Understanding geometric sequences allows students to identify and generate sequences easily by using the common ratio.

In any geometric sequence:

• The first term is denoted as \( a \).
• Each subsequent term is calculated by multiplying the previous term by the common ratio \( r \).
• In the sequence \( 1, -\dfrac{1}{2}, \dfrac{1}{4}, -\dfrac{1}{8}, \dots, -\dfrac{1}{128} \), every term is derived from its predecessor using \( r = -\dfrac{1}{2} \).

This sequence starts with 1 and alternates signs, while each term becomes half the previous one's absolute value. Thus, each term can be represented using the expression \( a \cdot r^{n-1} \), where \( a = 1 \) and \( r = -\dfrac{1}{2} \). Recognizing a sequence as geometric is key in tasks that involve finding specific terms or sums of series related to the sequence.