Power Series Expansion for Reciprocal of Cube of 1 + x/Proof 2
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Theorem
Let $x \in \R$ such that $-1 < x < 1$.
Then:
\(\ds \dfrac 1 {\paren {1 + x}^3}\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {k + 2} \paren {k + 1} } 2 x^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 - 3 x + 6 x^2 - 10 x^3 + 15 x^4 - \cdots\) |
Proof
\(\ds \frac 1 {\paren {1 + x} }\) | \(=\) | \(\ds \paren {1 + x}^{-3}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-3}^{\underline k} } {k!} x^k\) | General Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {\paren {-3} - j} } {k!} x^k\) | Definition of Falling Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {-\paren {j + 3} } } {\ds \prod_{j \mathop = 1}^k j} x^k\) | Definition of Factorial and simplifying | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 1}^k \paren {-\paren {j + 2} } } {\ds \prod_{j \mathop = 1}^k j} x^k\) | Translation of Index Variable of Product | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-1}^k \ds \prod_{j \mathop = 1}^k \paren {j + 2} } {\ds \prod_{j \mathop = 1}^k j} x^k\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\paren {k + 2} \paren {k + 1} } 2 x^k\) | simplification |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Binomial Series: $20.10$