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Chapter 7: Q. 44 (page 592)

In Exercises 43鈥�46 give the first five terms for a geometric sequence ${\{c r^{k}\}}_{k = 0}^{鈭�}$ with the specified values of $c a n d r .$
$c = - 2, r = - \frac{1}{3}$

Short Answer

Expert verified

The five terms are $- 2, \frac{2}{3}, - \frac{2}{9}, \frac{2}{27}, - \frac{2}{81} .$

Step by step solution

01

Step 1. Given information.

The given values are $c = - 2, r = - \frac{1}{3} .$

02

Step 2. First term.

The general sequence is $a_{k} = c r^{k} .$

On substituting the given values,

$a_{k} = (- 2) {(- \frac{1}{3})}^{k}$

For first term,

$k = 0, a_{0} = (- 2) {(- \frac{1}{3})}^{0} = - 2$

03

Step 3. Remaining terms.

Now, when,

$k = 1, a_{1} = (- 2) {(- \frac{1}{3})}^{1} = \frac{2}{3} k = 2, a_{2} = (- 2) {(- \frac{1}{3})}^{2} = - \frac{2}{9} k = 3, a_{3} = (- 2) {(- \frac{1}{3})}^{3} = \frac{2}{27} k = 4, a_{4} = (- 2) {(- \frac{1}{3})}^{4} = - \frac{2}{81}$

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Most popular questions from this chapter

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?

Prove Theorem 7.24 (a). That is, show that if c is a real number and ${鈭�}_{k = 1}^{鈭�} a_{k}$ is a convergent series, then ${鈭�}_{k = 1}^{鈭�} c a_{k} = c {鈭�}_{k = 1}^{鈭�} a_{k}$ .

Improper Integrals: Determine whether the following improper integrals converge or diverge.

${鈭�}_{1}^{鈭�} \frac{1}{x} d x .$

Prove that if ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges to L and ${鈭�}_{k = 1}^{鈭�} b_{k}$ converges to M , then the series ${鈭�}_{k = 1}^{鈭�} (a_{k} + b_{k}) = L + M$ .

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.

${鈭�}_{k = 3}^{鈭�} 鈥� \frac{1}{(k 鈭� 2)^{2}}$

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