Which of the following is true for the infinite series $\sum _ { n = 2 } ^ { \infty } \frac { 4 } { n ^ { 2 } - 1 } ?$
A)Its sequence of partial sums is not bounded from below.
B)Its sequence of partial sums is decreasing.
C) $\lim _ { n \rightarrow \infty } \frac { 4 } { n ^ { 2 } - 1 } \neq 0$
D)It is convergent.
E)It is divergent.

Question

Which of the following is true for the infinite series $\sum _ { n = 2 } ^ { \infty } \frac { 4 } { n ^ { 2 } - 1 } ?$ 
A)Its sequence of partial sums is not bounded from below.
B)Its sequence of partial sums is decreasing.
C) $\lim _ { n \rightarrow \infty } \frac { 4 } { n ^ { 2 } - 1 } \neq 0$
D)It is convergent.
E)It is divergent.

Quizplus · Accepted Answer

The series $\sum _ { n = 2 } ^ { \infty } \frac { 4 } { n ^ { 2 } - 1 }$ is a series of positive terms and can be rewritten using partial fraction decomposition as $\sum _ { n = 2 } ^ { \infty } \left( \frac { 2 } { n - 1 } - \frac { 2 } { n + 1 } \right)$, which shows it is a telescoping series. The terms cancel out in a way that leaves only a finite number of terms from the beginning and end when partial sums are considered, leading to a convergent series. Additionally, the general term of the series tends to 0 as $n$ approaches infinity, which is a necessary condition for convergence of an infinite series of positive terms.

Which of the Following Is True for the Infinite Series ∑n=2∞4n2−1?\sum _ { n = 2 } ^ { \infty } \frac { 4 } { n ^ { 2 } - 1 } ?∑n=2∞​n2−14​?

Which of the Following Is True for the Infinite Series $\sum _ { n = 2 } ^ { \infty } \frac { 4 } { n ^ { 2 } - 1 } ?$