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

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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

$\blacksquare$


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