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 |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.30$