Question

The sum $\sum _{r=2}^{\infty }\frac{1}{r^2-1}$ is equal to

• A. 1
• B. 3/4
• C. 4/3
• D. none

Answer 1

Answer: (B)

$S=\sum _{r=2}^{\infty }\frac{1}{r^2-1}=\sum _{r=2}^{\infty }\frac{1}{(r-1)(r+1)}$
$T_r=\frac{1}{(r-1)(r+1)}=\frac{1}{2}\left[\frac{1}{r-1}-\frac{1}{r+1}\right]\Rightarrow S_{\infty }=\frac{3}{4}$