Find the sum of the series $\sum _ { n = 3 } ^ { \infty } \frac { 1 } { 4 n ^ { 2 } - 1 }$ .
A) $\frac { 1 } { 5 }$
B) $\frac { 1 } { 20 }$
C) $\frac { 1 } { 2 }$
D) $\frac { 1 } { 10 }$
E) $\frac { 1 } { 4 }$
F) $\frac { 1 } { 3 }$
G) $\frac { 1 } { 6 }$
H)Divergent

Question

Find the sum of the series $\sum _ { n = 3 } ^ { \infty } \frac { 1 } { 4 n ^ { 2 } - 1 }$ .
A) $\frac { 1 } { 5 }$ 
B) $\frac { 1 } { 20 }$ 
C) $\frac { 1 } { 2 }$ 
D) $\frac { 1 } { 10 }$ 
E) $\frac { 1 } { 4 }$ 
F) $\frac { 1 } { 3 }$ 
G) $\frac { 1 } { 6 }$ 
H)Divergent

Quizplus · Accepted Answer

The series is telescoping. Rewrite the term as $\frac{1}{4n^2-1} = \frac{1}{(2n-1)(2n+1)}$ and use partial fraction decomposition: $\frac{1}{(2n-1)(2n+1)} = \frac{1}{2} \left( \frac{1}{2n-1} - \frac{1}{2n+1} \right)$. The series telescopes, and the sum is $\frac{1}{10}$.

Find the Sum of the Series ∑n=3∞14n2−1\sum _ { n = 3 } ^ { \infty } \frac { 1 } { 4 n ^ { 2 } - 1 }∑n=3∞​4n2−11​

Find the Sum of the Series $\sum _ { n = 3 } ^ { \infty } \frac { 1 } { 4 n ^ { 2 } - 1 }$