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Chapter 7: Q. 43 (page 592)

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

Short Answer

Expert verified

The first five terms are $3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \frac{3}{16}$ .

Step by step solution

01

Step 1. Given information.

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

02

Step 2. First term.

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

On substituting the given values,

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

For first term, $k = 0,$

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

03

Step 3. Remaining terms.

Now, when,

$k = 1, a_{1} = (3) {(\frac{1}{2})}^{'} = \frac{3}{2} k = 2, a_{2} = (3) {(\frac{1}{2})}^{2} = \frac{3}{4} k = 3, a_{3} = (3) {(\frac{1}{2})}^{3} = \frac{3}{8} k = 4, a_{4} = (3) {(\frac{1}{2})}^{4} = \frac{3}{16}$

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Most popular questions from this chapter

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{{(- 3)}^{k + 1}}{4^{k - 2}}$ converges or diverges. Give the sum of the convergent series.

True/False:

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: If $a_{k} 鈫� 0$ , then ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges.

(b) True or False: If ${鈭�}_{k = 1}^{鈭�} a_{k}$ converges, then $a_{k} 鈫� 0$ .

(c) True or False: The improper integral ${鈭�}_{1}^{鈭�} f (x) d x$ converges if and only if the series ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

(d) True or False: The harmonic series converges.

(e) True or False: If $p > 1$ , the series ${鈭�}_{k = 1}^{鈭�} k^{- p}$ converges.

(f) True or False: If $f (x) 鈫� 0$ as $x 鈫� 鈭�$ , then ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

(g) True or False: If ${鈭�}_{k = 1}^{鈭�} f (k)$ converges, then $f (x) 鈫� 0$ as $x 鈫� 鈭�$ .

(h) True or False: If ${鈭�}_{k = 1}^{鈭�} a_{k} = L$ and ${S_{n}}$ is the sequence of partial sums for the series, then the sequence of remainders ${L - S_{n}}$ converges to $0$ .

Use either the divergence test or the integral test to determine whether the series in Exercises 32鈥�43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

35. ${鈭�}_{k = 1}^{鈭�} \frac{k}{2 + k^{2}}$

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{2^{k + 2}}{5^{k - 1}}$ converges or diverges. Give the sum of the convergent series.

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

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