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Chapter 7: Q. 77 (page 605)

Prove that if $\{a_{k}\}$ is a sequence of nonzero terms with the property that $\lim_{k 鈫� 鈭�} a_{k} = 鈭�$ , then $\frac{1}{a_{k}} 鈫� 0$ .

Short Answer

Expert verified

The theorem has been proved.

Step by step solution

01

Step 1. Given Information.

The objective is to show that $\frac{1}{a_{k}} 鈫� 0$ .

02

Step 2. Proving the theorem.

It is given that $\lim_{k 鈫� 鈭�} a_{k} = 鈭�$ , therefore, for given $蔚 > 0$ , there exists a positive integer $N$ such that

$|a_{k}| > \frac{1}{蔚} f o r k > N . . . . . . . . . (1)$

For $k > N . |a_{k}| > \frac{1}{蔚}$ can be written as,

$\frac{1}{|a_{k}|} < 蔚$

Thus, for given, $蔚 > 0$ , there exists a positive integer $N$ such that

$\frac{1}{|a_{k}|} < 蔚$

Thus, $\frac{1}{a_{k}} 鈫� 0$ .

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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.

If $\lim_{k 鈫� 鈭�} \frac{a_{k}}{b_{k}} Where L i s$ a positive finite number, what may we conclude about the two series?

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 $({鈭�}_{k = 0}^{鈭�} c r^{k}) ({鈭�}_{k = 0}^{鈭�} b v^{k})$ ?

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.

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

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

${鈭�}_{k = 1}^{鈭�} 鈥� \frac{\tan^{鈭� 1} 鈦� k}{1 + k^{2}}$

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