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Chapter 7: Q, 36 (page 592)

In Exercises 31鈥�36 provide the first five terms of the given sequence. Unless specified, assume that the first term has index $1$ .
$\{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + 鈰� + \frac{1}{2^{k}}\}$

Short Answer

Expert verified

The first five terms are $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32} .$

Step by step solution

01

Step 1. Given information.

The given sequence is $\{\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + 鈰� + \frac{1}{2^{k}}\}$ .

02

Step 2. First term.

The general term of the sequence is $a_{k} = \frac{1}{2^{k}} .$

For first term, $k = 1,$

$a_{1} = \frac{1}{2^{1}}$

03

Step 3. Remaining terms.

Now, when,

$k = 2, a_{2} = \frac{1}{2^{2}} = \frac{1}{4} k = 3, a_{3} = \frac{1}{2^{3}} = \frac{1}{8} k = 4, a_{4} = \frac{1}{2^{4}} = \frac{1}{16} k = 5, a_{5} = \frac{1}{2^{5}} = \frac{1}{32}$

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Most popular questions from this chapter

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{k}{e^{k}}$

Let ${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent geometric series. If b and v are both nonzero, prove that ${鈭�}_{k = 0}^{鈭�} \frac{c r^{k}}{b v^{k}}$ is a geometric series. What condition(s) must be met for this series to converge?

Let ${鈭�}_{k = 1}^{鈭�} a_{k} b e a c o n v e r g e n t s e r i e s a n d {鈭�}_{k = 1}^{鈭�} b_{k} b e a d i v e r g e n t s e r i e s .$ Prove that the series ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k})$ diverges.

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$1.272727 . . .$

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