Which one or ones of the following conditions is necessary in order to apply the integral test to an infinite series? (5 points)
I. The series must have all positive terms.
II. The terms of the series must be decreasing.
III. The associated function must be continuous for the interval of the summation.
A) I only
B) I and II
C) II and III
D) I, II, and III
In order to apply the integral test to an infinite series, there are three necessary conditions that need to be met:
I. The series must have all positive terms.
II. The terms of the series must be decreasing.
III. The associated function must be continuous for the interval of the summation.
Let's go through each option to determine which ones satisfy these conditions:
A) I only: This option only satisfies condition I, but it doesn't fulfill the other two conditions. Therefore, option A is not the correct answer.
B) I and II: This option satisfies conditions I and II, which means the series has all positive terms and the terms are decreasing. However, it does not fulfill condition III. Therefore, option B is not the correct answer.
C) II and III: This option satisfies conditions II and III, which means the terms of the series are decreasing and the associated function is continuous for the interval of the summation. However, it does not fulfill condition I. Therefore, option C is not the correct answer.
D) I, II, and III: This option satisfies all three conditions. The series has all positive terms (condition I), the terms are decreasing (condition II), and the associated function is continuous for the interval of the summation (condition III). Therefore, option D is the correct answer.
Therefore, the correct answer is option D) I, II, and III.