Sum of Sequence of Products of 3 Consecutive Reciprocals/Corollary
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Theorem
- $\ds \sum_{j \mathop = 1}^\infty \frac 1 {j \paren {j + 1} \paren {j + 2} } = \frac 1 4$
Proof
\(\ds \sum_{j \mathop = 1}^\infty \frac 1 {j \paren {j + 1} \paren {j + 2} }\) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} \paren {j + 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \frac {n \paren {n + 3} } {4 \paren {n + 1} \paren {n + 2} }\) | Sum of Sequence of Products of 3 Consecutive Reciprocals | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \frac {1 + \frac 3 n} {4 \paren {1 + \frac 1 n} \paren {1 + \frac 2 n} }\) | dividing top and bottom by $n^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4\) | Basic Null Sequences |
$\blacksquare$