Q. 6 Let f(x) be a function that is c... [FREE SOLUTION]

Chapter 7: Q. 6 (page 624)

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, 鈭�)$ such that role="math" localid="1649081384626" $\underset{x 鈫� 鈭�}{l i m} f (x) = 伪 > 0$ . What can the divergence test tell us about the series ${鈭�}_{k = 1}^{鈭�} f (k)$ ?

Short Answer

Expert verified

By the divergence test, the series ${鈭�}_{k = 1}^{鈭�} f (k)$ is continuous, positive and decreasing on the interval $[1, 鈭�)$ with limit $\underset{x 鈫� 鈭�}{l i m} f (x) = 伪 > 0$ is divergent.

Step by step solution

Step 1. Given Information.

The function:

$\underset{x 鈫� 鈭�}{l i m} f (x) = 伪 > 0 o n [1, 鈭�)$ .

Step 2. Divergence test.

If the sequence ${a_{k}}$ does not converge to zero, then the series diverges.

$\underset{x 鈫� 鈭�}{l i m} f (x) = 伪 > 0$

Step 3. By divergent test.

the value of the limit is greater than zero, so by the divergent test series ${鈭�}_{k = 1}^{鈭�} f (k)$ is divergent.

So, by divergence test series, a function f(x) that is continuous, positive, and decreasing on the interval $[1, 鈭�)$ such that $\underset{x 鈫� 鈭�}{l i m} f (x) = 伪 > 0$ is divergent.

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91影视

Let f(x) be a function that is continuous, positive, and decreasing on the interval $[1, 鈭�)$ such that role="math" localid="1649081384626" $\underset{x 鈫� 鈭�}{l i m} f (x) = 伪 > 0$ . What can the divergence test tell us about the series ${鈭�}_{k = 1}^{鈭�} f (k)$ ?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Divergence test.

Step 3. By divergent test.

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91影视

Let f(x) be a function that is continuous, positive, and decreasing on the interval [1,鈭�)such that role="math" localid="1649081384626" limx鈫�鈭�f(x)=伪>0. What can the divergence test tell us about the series 鈭�k=1鈭�f(k)?

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Divergence test.

Step 3. By divergent test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Pure Maths

Mechanics Maths

Decision Maths

Applied Mathematics

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 $[1, 鈭�)$ such that role="math" localid="1649081384626" $\underset{x 鈫� 鈭�}{l i m} f (x) = 伪 > 0$ . What can the divergence test tell us about the series ${鈭�}_{k = 1}^{鈭�} f (k)$ ?