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Chapter 7: Q. 27 (page 592)

In Exercises, find a plausible formula for the general term of the given sequence.
$\{\frac{1}{3}, \frac{2}{9}, \frac{1}{9}, \frac{4}{81}, \frac{5}{243}, 鈥�\}$

Short Answer

Expert verified

The plausible formula for the general term of the sequence is $a_{k} = \frac{k}{3^{k}} .$

Step by step solution

01

Step 1. Given information.

The given sequence is $\{\frac{1}{3}, \frac{2}{9}, \frac{1}{9}, \frac{4}{81}, \frac{5}{243}, 鈥�\} .$

02

Step 2. plausible formula for the general term.

write the sequence in a pattern.

$\{\frac{1}{3}, \frac{2}{9}, \frac{1}{9}, \frac{4}{81}, \frac{5}{243}, 鈥�\} = \{\frac{1}{3^{1}}, \frac{2}{3^{2}}, \frac{3}{3^{3}}, \frac{4}{3^{4}}, \frac{5}{3^{5}}, 鈥�\} \{\frac{1}{3}, \frac{2}{9}, \frac{1}{9}, \frac{4}{81}, \frac{5}{243}, 鈥�\} = {\{\frac{k}{3^{k}}\}}_{k = 1}^{鈭�}$

here for every term, the numerator is the natural number k and the denominator is $3^{k} .$

So plausible formula for the general term of the sequence is $a_{k} = \frac{k}{3^{k}} .$

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Most popular questions from this chapter

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.

What is meant by a p-series?

In Exercises 48鈥�51 find all values of p so that the series converges.

${鈭�}_{k = 1}^{鈭�} \frac{k}{{(1 + k^{2})}^{p}}$

Consider the series

Fill in the blanks and select the correct word:

$I f \lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} = 鈭� and {鈭�}_{k = 1}^{鈭�}_____diverges t he n {鈭�}_{k = 1}^{鈭�}_____(converges / diverges) .$

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.

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

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