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 .
C is for sure false; did you try out the others?
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 C or D.
A only true
To determine which of the statements is true, let's analyze each statement separately.
A) The nth term test can never be used to show that a series converges.
The nth term test, also known as the divergence test, states that if the limit as n approaches infinity of the nth term of a series is not zero, then the series diverges. Thus, statement A is true.
B) The integral test can be applied to a series even if all the terms are not positive.
The integral test allows you to determine the convergence or divergence of a series by comparing it to the convergence or divergence of an integral. This test can be applied as long as the function being integrated is positive, continuous, and decreasing. Therefore, statement B is false, as the terms must be positive for the integral test to be applicable.
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.
The series in question is known as the p-series. The p-series converges if the value of p is less than 1 and diverges if p is greater than or equal to 1. Therefore, statement C is true.
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.
The Integral Test states that if a series and a corresponding integral satisfy certain conditions, then they both converge or diverge. However, statement D is not true because it states that the series is equal to the integral, which is incorrect. The Integral Test only establishes convergence or divergence; it does not give the exact value.
Therefore, the correct statement is:
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.