Which of the following statements is true? (4 points)
A) The nth term test can never be used to show that a series converges.
B) The integral test can be applied to a series even if all the terms are not positive.
C) The series the summation from n equals 1 to infinity of 1 over n raised to the p power converges if p < 1 and diverges if p > 1.
D) If an and f(x) satisfy the requirements of the Integral Test, and if the integral from 1 to infinity of f of x, dx converges, then the summation from n equals 1 to infinity of a sub n equals the integral from 1 to infinity of f of x, dx .
YOU: C is for sure false; did you try out the others?
Me: Are you sure? I tried the others and all them were false except C and D. I am not sure though, you might be right? Is it A or D.
Based on your statement that C is false, and the fact that you are unsure about A or D, we can conclude that the correct statement is D) If an and f(x) satisfy the requirements of the Integral Test, and if the integral from 1 to infinity of f(x)dx converges, then the summation from n equals 1 to infinity of a sub n equals the integral from 1 to infinity of f(x)dx.
To determine which statement is true, let's go through each option and consider the relevant concepts.
A) The nth term test, also known as the divergence test, states that if the limit of the nth term of a series as n approaches infinity is not zero, then the series diverges. However, it does not provide any information about convergence. Therefore, statement A is true.
B) The integral test states that if a series satisfies the conditions of the integral test, then the convergence or divergence of the series can be determined by analyzing the behavior of a related improper integral. This test can be applied even if all the terms of the series are not positive. Therefore, statement B is true.
C) The statement is about the p-series, which is a special type of series in the form of Σ(1/n^p) where n starts from 1 and goes to infinity. The p-series converges if p > 1 and diverges if p <= 1. Therefore, statement C is true.
D) The statement is known as the Integral Test for series convergence. It states that if a function f(x) satisfies certain requirements and its corresponding integral converges, then the series of its terms also converges. Therefore, statement D is true.
Based on this analysis, statements A, B, C, and D are all true.
C is the Riemann zeta function ζ(p)
you can read about it online