Chapter 7: Q. 8 (page 624)

Explain how you could adapt the integral test to analyze a series $\sum_{k = 1}^{\infty} f (k)$ in which the function $f : [1, \infty) \to ℝ$ is continuous, negative, and increasing.

Short Answer

Expert verified

By the integral test, the series $\sum_{k = 1}^{\infty} f (k)$ is divergent.

Step by step solution

Step 1. Given Information.

The series:

$\sum_{k = 1}^{\infty} f (k)$

Step 2. Integral test.

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

$\int_{1}^{\infty} - f (x) d x = \underset{k \to \infty}{l i m} \int_{1}^{k} - α . d x = \underset{k \to \infty}{l i m} {[- α x]}_{1}^{k} = \underset{k \to \infty}{l i m} [- α k + α] = 0$

Step 3. Convergent or divergent.

By the integral test, the improper integral $\int_{1}^{\infty} f (x) d x$ is divergent.

So the series $\sum_{k = 1}^{\infty} f (k)$ is divergent.

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Explain how you could adapt the integral test to analyze a series $\sum_{k = 1}^{\infty} f (k)$ in which the function $f : [1, \infty) \to ℝ$ is continuous, negative, and increasing.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Integral test.

Step 3. Convergent or divergent.

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Explain how you could adapt the integral test to analyze a series ∑k=1∞f(k)in which the functionf:[1,∞)→ℝ is continuous, negative, and increasing.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Integral test.

Step 3. Convergent or divergent.

One App. One Place for Learning.

Most popular questions from this chapter

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Explain how you could adapt the integral test to analyze a series $\sum_{k = 1}^{\infty} f (k)$ in which the function $f : [1, \infty) \to ℝ$ is continuous, negative, and increasing.