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Chapter 5: Problem 7

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

Short Answer

Expert verified
b. \(\sum_{k=0}^{5} 2^{k}\)

Step by step solution

01

Understand the Series

The series given is: \(1 + 2 + 4 + 8 + 16 + 32\). Each term is obtained by multiplying the previous term by 2.
02

Recognize the Pattern

Notice the pattern in the series: \(1, 2, 4, 8, 16, 32\). This can be expressed as \(2^0, 2^1, 2^2, 2^3, 2^4, 2^5\). The powers of 2 start from 0 and go to 5.
03

Convert to Sigma Notation

To express this series using sigma notation, we need to use exponents of 2, starting from \(k = 0\) and ending at \(k = 5\). Therefore, the sigma notation is: \(\sum_{k=0}^{5} 2^k\).
04

Compare with Options

Now, let's compare this sigma notation with the provided options. - Option a: \(\sum_{k=1}^{6} 2^{k-1}\) does not match exactly as the starting power is different.- Option b: \(\sum_{k=0}^{5} 2^k\) matches perfectly.- Option c: \(\sum_{k=1}^{4} 2^{k+1}\) does not match as the range is limited to 4 terms.

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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 series 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. In our exercise, the series is given as \(1 + 2 + 4 + 8 + 16 + 32\). If you examine these numbers closely, you'll see that each term is twice the term before it. Here, the common ratio is 2.
  • The first term is 1.
  • Each subsequent term is multiplied by 2, so \(1 \cdot 2 = 2\), \(2 \cdot 2 = 4\), and so forth.
This particular sequence fits perfectly with the characteristics of a geometric series. The importance of recognizing this pattern lies in how it allows us to express the series using sigma notation effectively.
Exponents
Exponents play a critical role in expressing geometric series compactly. When dealing with geometric series, exponents help to show the repeated multiplication of the common ratio. For instance, looking at the series \(1 + 2 + 4 + 8 + 16 + 32\), you can rewrite it using powers of 2:
  • \(1 = 2^0\)
  • \(2 = 2^1\)
  • \(4 = 2^2\)
  • \(8 = 2^3\)
  • \(16 = 2^4\)
  • \(32 = 2^5\)
This transformation simplifies our understanding by showing how each term relates to the previous terms as powers of 2. The transition from numerical multiplication to exponentiation paves the way for using sigma notation to see the series as a whole.
Series Representation
Representation of a series using sigma notation allows us to write long series in a compact form. Sigma notation simplifies conveying the entire set of operations and their scope. For the series \(1 + 2 + 4 + 8 + 16 + 32\), once we recognize it as a geometric series with terms \(2^0\) to \(2^5\), we use sigma notation as follows:
  • The expression \(\sum_{k=0}^{5} 2^k\) clearly describes the series with all terms articulated through the index \(k\).
  • "\(k = 0\)" indicates the starting index, and "\(5\)" the ending index, covering all terms from \(2^0\) to \(2^5\).
Opt for option b, \(\sum_{k=0}^{5} 2^k\), over the others as it fully represents the given series. Understanding series representation through sigma notation is a powerful tool for managing such series efficiently.

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Most popular questions from this chapter

Use a CAS to perform the following steps: a. Plot the functions over the given interval. b. Partition the interval into \(n=100,200,\) and 1000 subintervals of equal length, and evaluate the function at the midpoint of each subinterval. c. Compute the average value of the function values generated in part (b). d. Solve the equation \(f(x)=\) (average value) for \(x\) using the average value calculated in part (c) for the \(n=1000\) partitioning. $$f(x)=x \sin \frac{1}{x} \quad \text { on } \quad\left[\frac{\pi}{4}, \pi\right]$$

Find the total area between the region and the \(x\) -axis. $$y=x^{1 / 3}-x, \quad-1 \leq x \leq 8.$$

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Suppose that \(f\) has a positive derivative for all values of \(x\) and that \(f(1)=0 .\) Which of the following statements must be true of the function $$g(x)=\int_{0}^{x} f(t) d t ?$$. Give reasons for your answers. a. \(g\) is a differentiable function of \(x\) b. \(g\) is a continuous function of \(x\). c. The graph of \(g\) has a horizontal tangent at \(x=1\) d. \(g\) has a local maximum at \(x=1\) e. \(g\) has a local minimum at \(x=1\) f. The graph of \(g\) has an inflection point at \(x=1\) g. The graph of \(d g / d x\) crosses the \(x\) -axis at \(x=1\)

Each of the following functions solves one of the initial value.Which function solves which problem? Give brief reasons for your answers. a. \(y=\int_{1}^{x} \frac{1}{t} d t-3\) b. \(y=\int_{0}^{x} \sec t d t+4\) c. \(y=\int_{-1}^{x} \sec t d t+4\) d. \(y=\int_{\pi}^{x} \frac{1}{t} d t-3\) $$\frac{d y}{d x}=\frac{1}{x}, \quad y(\pi)=-3.$$
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