Sum of Sequence of Products of Squares of 3 Consecutive Reciprocals/Proof 1
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Theorem
\(\ds \sum_{j \mathop = 1}^\infty \frac 1 {j^2 \paren {j + 1}^2 \paren {j + 2}^2}\) | \(=\) | \(\ds \frac 1 {1^2 \times 2^2 \times 3^2} + \frac 1 {2^2 \times 3^2 \times 4^2} + \frac 1 {3^2 \times 4^2 \times 5^2} + \frac 1 {4^2 \times 5^2 \times 6^2} + \cdots\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {4 \pi^2 - 39} {16}\) |
Proof
We use partial fraction expansion to expand $\dfrac 1 {j^2 \paren {j + 1}^2 \paren {j + 2}^2}$.
Let $\dfrac 1 {j^2 \paren {j + 1}^2 \paren {j + 2}^2} = \dfrac A j + \dfrac B {j^2} + \dfrac C {j + 1} + \dfrac D {\paren {j + 1}^2} + \dfrac E {j + 2} + \dfrac F {\paren {j + 2}^2}$.
This has solutions $A = - \dfrac 3 4, B = \dfrac 1 4, C = 0, D = 1, E = \dfrac 3 4, F = \dfrac 1 4$.
Thus:
\(\ds \sum_{j \mathop = 1}^\infty \frac 1 {j^2 \paren {j + 1}^2 \paren {j + 2}^2}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \paren {-\frac 3 {4j} + \frac 1 {4 j^2} + \frac 1 {\paren {j + 1}^2} + \frac 3 {4 \paren {j + 2} } + \frac 1 {4 \paren {j + 2}^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \paren {\frac 1 {4 j^2} + \frac 1 {\paren {j + 1}^2} + \frac {3 j - 3 \paren {j + 2} } {4 j \paren {j + 2} } + \frac 1 {4 \paren {j + 2}^2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{j \mathop = 1}^\infty \frac 1 {4 j^2} + \sum_{j \mathop = 1}^\infty \frac 1 {\paren {j + 1}^2} + \sum_{j \mathop = 1}^\infty \frac 1 {4 \paren {j + 2}^2} - \sum_{j \mathop = 1}^\infty \frac 6 {4 j \paren {j + 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 4 \sum_{j \mathop = 1}^\infty \frac 1 {j^2} + \sum_{j \mathop = 2}^\infty \frac 1 {j^2} + \frac 1 4 \sum_{j \mathop = 3}^\infty \frac 1 {j^2} - \frac 3 2 \sum_{j \mathop = 1}^\infty \frac 1 {j \paren {j + 2} }\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 3 2 \sum_{j \mathop = 1}^\infty \frac 1 {j^2} - 1 - \frac 1 4 - \frac 1 {16} - \frac 3 2 \paren {\frac 3 4}\) | Corollary to Sum of Sequence of Products of Consecutive Odd and Consecutive Even Reciprocals | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 3 2 \paren {\frac {\pi^2} 6} - \frac {21} {16} - \frac 9 8\) | Basel Problem | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {4 \pi^2 - 39} {16}\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 19$: Series involving Reciprocals of Powers of Positive Integers: $19.33$