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Chapter 7: Q. 6 (page 639)

Use Exercise 5 to explain why the ratio test will be inconclusive for every series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which $a_{k}$ is a rational function of $k$ .

Short Answer

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Step by step solution

01

Step 1. Given Information.

The given series is ${鈭�}_{k = 1}^{鈭�} a_{k}$ .

02

Step 2. Explanation.

Let $r (x) = \frac{a_{0}}{b_{0}} x + \frac{a_{1}}{b_{1}} x + \frac{a_{2}}{b_{2}} x^{2} + 鈥� + \frac{a_{n}}{b_{n}} x^{n} + 鈥� with b_{i} 鈮� 0 = r (x) = {鈭�}_{i = 0}^{鈭�} \frac{a_{i}}{b_{i}} x^{i} r (x + 1) = {鈭�}_{i = 0}^{鈭�} \frac{a_{i}}{b_{i}} (x + 1)^{i} \lim_{x 鈫� 鈭�} \frac{r (x + 1)}{r (x)} = \lim_{x 鈫� 鈭�} {(1 + \frac{1}{x})}^{'} = 1$

That's why the ratio test becomes inconclusive for the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which $a_{k}$ is the rational function of $k$ .

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Most popular questions from this chapter

Provide a more general statement of the integral test in which the function f is continuous and eventually positive, and decreasing. Explain why your statement is valid.

Whenever a certain ball is dropped, it always rebounds to a height p% (0 < p < 100) of its original position. What is the total distance the ball travels before coming to rest when it is dropped from a height of h meters?

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 $0$ .

Determine whether the series ${鈭�}_{k = 2}^{鈭�} {(\frac{- 3}{5})}^{k}$ converges or diverges. Give the sum of the convergent series.

Explain why, if n is an integer greater than 1, the series ${鈭�}_{k = 1}^{鈭�} \frac{1}{\sqrt[n]{k}}$ diverges.

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