# Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals/Corollary

## Corollary to Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals

 $\ds \sum_{j \mathop = 1}^\infty \frac 1 {j \left({j + 2}\right)}$ $=$ $\ds \frac 1 {1 \times 3} + \frac 1 {2 \times 4} + \frac 1 {3 \times 5} + \frac 1 {4 \times 6} + \cdots$ $\ds$ $=$ $\ds \frac 3 4$

## Proof

 $\ds \sum_{j \mathop = 1}^\infty \frac 1 {j \left({j + 2}\right)}$ $=$ $\ds \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \left({j + 2}\right)}$ $\ds$ $=$ $\ds \lim_{n \mathop \to \infty} \left({\frac 3 4 - \frac {2 n + 3} {2 \left({n + 1}\right) \left({n + 2}\right)} }\right)$ Sum of Sequence of Products of Consecutive Odd Reciprocals $\ds$ $=$ $\ds \frac 3 4 - \lim_{n \mathop \to \infty} \frac {\frac 2 n + \frac 3 {n^2} } {2 \left({1 + \frac 1 n}\right) \left({1 + \frac 2 n}\right)}$ dividing top and bottom by $n^2$ $\ds$ $=$ $\ds \frac 3 4$ Basic Null Sequences

$\blacksquare$