Sum of Sequence of Products of 3 Consecutive Reciprocals/Corollary

Theorem

$\ds \sum_{j \mathop = 1}^\infty \frac 1 {j \paren {j + 1} \paren {j + 2} } = \frac 1 4$

$\ds \sum_{j \mathop = 1}^\infty \frac 1 {j \paren {j + 1} \paren {j + 2} }$

$=$

$\ds \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} \paren {j + 2} }$

$\ds $

$=$

$\ds \lim_{n \mathop \to \infty} \frac {n \paren {n + 3} } {4 \paren {n + 1} \paren {n + 2} }$

$\ds $

$=$

$\ds \lim_{n \mathop \to \infty} \frac {1 + \frac 3 n} {4 \paren {1 + \frac 1 n} \paren {1 + \frac 2 n} }$

dividing top and bottom by $n^2$

$\ds $

$=$

$\ds \frac 1 4$

$\blacksquare$