Problem 8 - Thomas Calculus

Question

Which of the following express \(1-2+4-8+16-32\) in sigma notation? $$ \text { a. }\sum_{k=1}^{6}(-2)^{k-1} \quad \text { b. } \sum_{k=0}^{5}(-1)^{k} 2^{k} \quad \text { c. } \sum_{k=-2}^{3}(-1)^{k+1} 2^{k+2} $$

Step-by-Step Solution

Answer

Option b: \(\sum_{k=0}^{5} (-1)^{k} 2^{k}\)

Step 1: Recognize the Series Pattern

The series given is \(1 - 2 + 4 - 8 + 16 - 32\). We need to identify the rule that dictates the successive terms. Observing the series, we notice that each term alternates in sign and is a power of 2.

Step 2: Analyze the Pattern and Formula

The series can be written as \((-1)^k \cdot 2^k\), where \(k\) takes on integer values starting with 0. For example, \((-1)^0 \cdot 2^0 = 1\), \((-1)^1 \cdot 2^1 = -2\), and so forth.

Step 3: Write the Series as a Sigma Notation

Based on the pattern, the terms of the series can be expressed as \((-1)^k 2^k\). We will sum these terms from \(k = 0\) to \(k = 5\) to include the six terms of the series: \(1, -2, 4, -8, 16, -32\). Thus, the sigma notation is \( \sum_{k=0}^{5} (-1)^k 2^k\).

Step 4: Match with the Provided Options

Among the given choices, option b matches our sigma notation derivation: \(\sum_{k=0}^{5} (-1)^{k} 2^{k} \). This confirms that it's the correct representation of the series in sigma notation.

Key Concepts

Series Patterns

A series pattern is a sequence of numbers that follows a particular set of rules or formulas, believed to be important for understanding larger mathematical concepts. When you analyze a series like \(1 - 2 + 4 - 8 + 16 - 32\), it is crucial to identify how each number is related to the next. In this case, each term is a power of 2. This means that each number in the sequence can be expressed as \(2^n\) for some whole number \(n\).
Not only is the absolute value a power of 2, but also the signs are alternating between positive and negative. Understanding the pattern allows you to write equations that generalize the series with a fixed formula, prime for sigma notation. Recognizing series patterns simplifies identifying formulas and helps in calculations of sums over extensive sequences.

Alternating Series

Alternating series are fascinating because they switch sign with each consecutive term. This kind of series features prominently in the given sequence \(1 - 2 + 4 - 8 + 16 - 32\). In mathematics, alternating series can be written using powers of \((-1)\), a number cycling continuously between positive and negative.
For our series, this alternating nature can be represented as \((-1)^k\), where \(k\) is an integer counting term positions. If \(k\) is even, then \((-1)^k = 1\), giving a positive term. If \(k\) is odd, then \((-1)^k = -1\), resulting in a negative term. Understanding this is important in correctly expressing our original series in sigma notation, capturing both the numerical and sign pattern in a formulaic expression.

Exponents in Series

Exponents in a series are key, especially in sequences involving repeated multiplication. Observe that each term of the given series \(1 - 2 + 4 - 8 + 16 - 32\) can be reframed using exponents: \(2^0, 2^1, 2^2, 2^3, etc.\) This exponential nature indicates rapid growth or decrease of terms in the series.
Understanding exponents allows you to rewrite series in a more manageable form. For instance, a term \(2^k\) represents how many times you multiply the base, which is 2 in this example. This comprehension allows you to reformulate sums and express complex ideas concisely using sigma notation and other forms. Such knowledge is valuable in calculus, algebra, and beyond, where evaluating series quickly and effectively makes problem-solving more efficient.