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Chapter 7: Q. 7 (page 624)

Let f(x) be a function that is continuous, positive, and decreasing on the interval [1,鈭�)such that limx鈫�鈭�f(x)=伪>0, What can the integral tells us about the series鈭�k=1鈭�f(k) ?

Short Answer

Expert verified

The series 鈭�k=1鈭�f(k) is divergent.

Step by step solution

01

Step 1. Given Information.

The function:

limx鈫�鈭�f(x)=伪>0on[1,鈭�)

02

Step 2. Integral test.

By the integral test, the function will both converge or diverge.

鈭�1鈭�f(x)dx=limk鈫�鈭�鈭�1k伪.dx=limk鈫�鈭�[伪x]1k=limk鈫�鈭�[伪k-伪]=0

03

Step 3. Convergent or divergent.

By the integral test, the improper integral 鈭�1鈭�f(x)dxis divergent.

So the series 鈭�k=1鈭�f(k) is divergent.

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Most popular questions from this chapter

Let a:[1,鈭�)鈫�鈩�be a continuous, positive, and decreasing function. Complete the proof of the integral test (Theorem 7.28) by showing that if the improper integral 鈭�1鈭�a(x)dxconverges, then the series localid="1649180069308" 鈭�k=1鈭�a(k)does too.

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.

0.99999...

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

鈭�k=1鈭�1(c+k)p,wherec>0

Determine whether the series 鈭�k=0鈭�e蟺3kconverges or diverges. Give the sum of the convergent series.
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