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Chapter 5: Problem 8

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

Short Answer

Expert verified
b: \(\sum_{k=0}^{5}(-1)^{k} 2^{k}\) is the correct sigma notation.

Step by step solution

01

Identify the sequence pattern

The given expression is \(1 - 2 + 4 - 8 + 16 - 32\). Observe the sequence carefully: the numbers alternate signs and are powers of 2. This can be expressed as \((-1)^n \cdot 2^n\) for each term, starting from \(n = 0\) (i.e., \(1\) which is \(2^0\) with a positive sign).
02

Translate to sigma notation

To convert the sequence \(1, -2, 4, -8, 16, -32\) to sigma notation, observe that there are 6 terms indexed from 0 to 5: \(1 = (-1)^0 \cdot 2^0, -2 = (-1)^1 \cdot 2^1, \ldots, -32 = (-1)^5 \cdot 2^5\). The pattern can be written as \(\sum_{k=0}^{5}(-1)^{k} 2^{k}\).
03

Select the correct option

Comparing the derived sigma notation \(\sum_{k=0}^{5}(-1)^{k} 2^{k}\) with the given choices, the correct answer is choice b.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Geometric Series
A geometric series is a mathematical concept where the terms in the series form a sequence with a constant ratio between consecutive terms. In our exercise, the sequence follows the rule of doubling or halving each previous term while alternating in sign. This can be formulated through a sigma notation.
  • In a geometric series, each term is a fixed multiple (common ratio) of the previous term, which makes it predictable and relatively easy to work with.
  • The alternating nature in the expression "1 - 2 + 4 - 8 + 16 - 32" indicates a geometric sequence where the absolute value of each term is multiplied by 2.
  • The common ratio here is -2, which signifies each term is transformed by multiplying by 2 and changing its sign.
These properties make the geometric series a cornerstone in understanding sigma notation, allowing students to map series with concise mathematical expressions.
Alternating Series
An alternating series is characterized by its consecutive terms changing sign, as seen in the given sequence: "1 - 2 + 4 - 8 + 16 - 32".
  • The alternating effect is often represented using \((-1)^k\), where \(k\) is an integer that goes from one term to the next.
  • This pattern of sign changes distinguishes an alternating series from non-alternating sequences, where every term would have the same sign.
  • In the sequence provided, \( (-1)^k \) effectively flips the sign from positive to negative for every new term, which can be essential in series representation.
The telegraphic shift of signs in alternating series brings forth interesting solutions when using sigma notation, emphasizing how signs can be systematically managed.
Exponents
Exponents are a fundamental concept in this exercise, as each term in the series is a power of 2. This concept is crucial in the formation and understanding of geometric and alternating series.
  • In mathematics, exponents are used to express repeated multiplication compactly. For example, in the sequence 1, -2, 4, -8, 16, -32, the term 16 can be written as 2^4.
  • Each number is generated by raising 2 to an integer power, starting from 2^0 for the first term.
  • The exponent increases by one for each successive term, maintaining a consistent pattern of power increase in the sequence.
  • Understanding exponents allows us to see the growth or decay trends inherent in series, providing clear frameworks for evaluating and writing them in sigma notation.
Exponents are pivotal in mathematics, building bridges between abstract series and practical computation for sequences like those described in the exercise.

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