Sum of Sequence of Products of Consecutive Odd Reciprocals/Corollary

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Corollary to Sum of Sequence of Products of Consecutive Odd Reciprocals

\(\ds \sum_{j \mathop = 0}^\infty \frac 1 {\paren {2 j + 1} \paren {2 j + 3} }\) \(=\) \(\ds \frac 1 {1 \times 3} + \frac 1 {3 \times 5} + \frac 1 {5 \times 7} + \frac 1 {7 \times 9} + \cdots\)
\(\ds \) \(=\) \(\ds \frac 1 2\)


Proof

\(\ds \sum_{j \mathop = 0}^\infty \frac 1 {\paren {2 j + 1} \paren {2 j + 3} }\) \(=\) \(\ds \lim_{n \mathop \to \infty} \sum_{j \mathop = 0}^n \frac 1 {\paren {2 j + 1} \paren {2 j + 3} }\)
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \frac {n + 1} {2 n + 3}\) Sum of Sequence of Products of Consecutive Odd Reciprocals
\(\ds \) \(=\) \(\ds \lim_{n \mathop \to \infty} \frac {1 + \frac 1 n} {2 + \frac 3 n}\) dividing top and bottom by $n$
\(\ds \) \(=\) \(\ds \frac 1 2\) Basic Null Sequences

$\blacksquare$


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