Use Theorem 9.11 to determine the convergence or divergence of the series.
A) converges
B) diverges
C) Theorem 9.11 inconclusive

Question

Use Theorem 9.11 to determine the convergence or divergence of the series. 
A) converges 
B) diverges 
C) Theorem 9.11 inconclusive

Quizplus · Accepted Answer

Since Theorem 9.11 states that a series diverges if the limit of the terms does not approach zero, we can investigate whether the terms of the series approach zero or not. Notice that the nth term of the series is given by (n^2 + 1)/(n^3 + 1)^2. As n approaches infinity, both the numerator and denominator approach infinity. However, since the denominator has a higher degree, the fraction approaches zero. Thus the limit of the terms is zero, which does not allow us to determine whether the series converges or diverges. Therefore, Theorem 9.11 is inconclusive, and it is possible that the series may converge or diverge. However, we can use other tests or methods to determine its convergence or divergence.