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Chapter 7: Q. 9 (page 639)

Let ${鈭�}_{k = 1}^{鈭�} a_{k}$ be a series in which all the terms are positive. If $\lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} > 1$ , explain why both the ratio test and the divergence test could be used to show that the series diverges .

Short Answer

Expert verified

Hence proved.

Step by step solution

01

Step 1. Given information.

We are given,

$\lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} > 1$

02

Step 2. Ratio Test.

The ratio test for ${鈭�}_{k = 1}^{鈭�} a_{k} s e r i e s a n d L = \lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}}$ is given by,

$1. If L < 1 series converges. 2. If L > 1 series diverges. 3. If L = 1 the test is inconclusive. Since, \lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} > 1, therefore, L > 1 .$

Hence, it diverges.

03

Step 2. Divergence Test.

We know,

According to the Divergence Test if the sequence $\{a_{k}\}$ does not converge to zero, then the series $\lim_{k 鈫� 鈭�} \frac{a_{k + 1}}{a_{k}} > 1$ diverges.

Since $\lim_{k 鈫� 鈭�} a_{k} 鈮� 0$

Here, $a_{k}$ does not converge to 0 .

Hence, the series ${鈭�}_{k = 1}^{鈭�} a_{k}$ diverges by Divergence Test.

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Most popular questions from this chapter

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

${鈭�}_{k = 1}^{鈭�} \frac{1}{{(c + k)}^{p}}, w h e r e c > 0$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which $a_{k} 鈫� 0$ .

(b) A divergent p-series.

(c) A convergent p-series.

Determine whether the series ${鈭�}_{k = 0}^{鈭�} 蟺 {(\frac{e}{3})}^{k}$ converges or diverges. Give the sum of the convergent series.

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

${鈭�}_{1}^{鈭�} \frac{1}{x} d x .$

Let ${鈭�}_{k = 0}^{鈭�} c r^{k}$ and ${鈭�}_{k = 0}^{鈭�} b v^{k}$ be two convergent geometric series. Prove that ${鈭�}_{k = 0}^{鈭�} (c r^{k} . b v^{k})$ converges. If neither c nor b is 0, could the series be $({鈭�}_{k = 0}^{鈭�} c r^{k}) ({鈭�}_{k = 0}^{鈭�} b v^{k})$ ?

See all solutions

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