Chapter 7: Q. 7 (page 624)

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, \infty)$ such that $\underset{x \to \infty}{l i m} f (x) = α > 0$ , What can the integral tells us about the series $\sum_{k = 1}^{\infty} f (k)$ ?

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information.

The function:

$\underset{x \to \infty}{l i m} f (x) = α > 0 o n [1, \infty)$

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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Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, \infty)$ such that $\underset{x \to \infty}{l i m} f (x) = α > 0$ , What can the integral tells us about the series $\sum_{k = 1}^{\infty} f (k)$ ?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Integral test.

Step 3. Convergent or divergent.

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

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

Recommended explanations on Math Textbooks

Applied Mathematics

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, \infty)$ such that $\underset{x \to \infty}{l i m} f (x) = α > 0$ , What can the integral tells us about the series $\sum_{k = 1}^{\infty} f (k)$ ?