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Chapter 7: Q. 9 (page 624)

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.

Short Answer

Expert verified

The statement is valid because the tail of the series determines the convergence or divergence.

Step by step solution

01

Step 1. Given Information.

The function f.

02

Step 2. The integral test.

If $f (x) : [1, \infty) \to ℝ$ is continuous, eventually positive and decreasing on $[1, \infty)$ , and $f_{k}$ is the sequence defined by $f_{k} = {f (k)}$ for every $k \in ℤ^{+}$ , then $\sum_{k = 1}^{\infty} f_{k} a n d \int_{1}^{\infty} f (x) d x$ either both converge or diverge.

03

Step 3. To explain.

The statement is valid because the tail of the series determines the convergence or divergence.

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Most popular questions from this chapter

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

$\sum_{k = 2}^{\infty} \frac{1}{k {(\ln (k))}^{p}}$

What is meant by the remainder $R_{n}$ of a series $\sum_{k = 1}^{\infty} a_{k}$

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.

$\sum_{k = 1}^{\infty} \frac{1}{k}$

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.

$\sum_{k = 3}^{\infty} \frac{1}{(k - 2)^{2}}$

Determine whether the series $\sum_{j = 2}^{\infty} \frac{2^{j}}{3}$ converges or diverges. Give the sum of the convergent series.

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