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Chapter 7: Q. 15 (page 624)

Find an example of a continuous function f : $[1, 鈭�) 鈫� R$ such that ${鈭�}_{1}^{鈭�} f (x) d x$ diverges and localid="1649077247585" ${鈭�}_{k = 1}^{鈭�} f (k)$ converges.

Short Answer

Expert verified

An example is $\sin (蟺虫)$ .

Step by step solution

01

Step 1. Given information.

Consider the given question,

The function is $[1, 鈭�) 鈫� R$ .

The given integral is ${鈭�}_{1}^{鈭�} f (x) d x$ .

02

Step 2. Evaluate the integral.

Evaluating the integral,

${鈭�}_{1}^{鈭�} f (x) d x = {鈭�}_{1}^{鈭�} \sin (蟺虫) d x = \lim_{k 鈫� 鈭�} {鈭�}_{1}^{k} \sin (蟺虫) d x = \lim_{k 鈫� 鈭�} {鈭�}_{1}^{k} {[\frac{- \cos (蟺虫)}{蟺}]}^{k}_{1} = \lim_{k 鈫� 鈭�} {鈭�}_{1}^{k} [\frac{- \cos (蟺虫) + \cos 蟺}{蟺}] = \lim_{k 鈫� 鈭�} {鈭�}_{1}^{k} [\frac{- \cos (蟺虫) - 1}{蟺}] = 鈭�$

Thus, the improper integral ${鈭�}_{1}^{鈭�} \sin (蟺虫) d x$ is divergent.

03

Step 3. To prove the series as convergent.

The value of ${鈭�}_{k = 1}^{鈭�} f (k) = {鈭�}_{k = 1}^{鈭�} \sin (蟺 k)$ is given below,

${鈭�}_{k = 1}^{鈭�} \sin 蟺办 = \sin 蟺 + \sin 2 蟺 + \sin 3 蟺 + . . . {鈭�}_{k = 1}^{鈭�} \sin 蟺办 = 0 + 0 + 0 + . . . {鈭�}_{k = 1}^{鈭�} \sin 蟺办 = = 0$

Thus, the series ${鈭�}_{k = 1}^{鈭�} f (k) = {鈭�}_{k = 1}^{鈭�} \sin (蟺 k)$ is convergent.

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Most popular questions from this chapter

Find the values of x for which the series ${鈭�}_{k = 0}^{鈭�} x^{k}$ converges.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

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

Determine whether the series $\frac{99}{40} - \frac{99}{20} + \frac{99}{10} - \frac{99}{5} + 鈥�$ converges or diverges. Give the sum of the convergent series.

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$3.454545 . . .$

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.237237237 . . .$

See all solutions

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