Power Series Expansion for Reciprocal of Square Root of 1 - x
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Theorem
Let $x \in \R$ such that $-1 < x \le 1$.
Then:
| \(\ds \dfrac 1 {\sqrt {1 - x} }\) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {2 k}!} {\paren {2^k k!}^2} x^k\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^3 + \cdots\) |
Proof 1
| \(\ds \frac 1 {\sqrt {1 + x} }\) | \(=\) | \(\ds 1 - \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 - \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^3 + \cdots\) | Power Series Expansion for $\dfrac 1 {\sqrt {1 + x} }$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \frac 1 {\sqrt {1 - x} }\) | \(=\) | \(\ds 1 - \frac 1 2 \paren {-x} + \frac {1 \times 3} {2 \times 4} \paren {-x}^2 - \frac {1 \times 3 \times 5} {2 \times 4 \times 6} \paren {-x}^3 + \cdots\) | substituting $x \gets -x$ | ||||||||||
| \(\ds \) | \(=\) | \(\ds 1 + \frac 1 2 x + \frac {1 \times 3} {2 \times 4} x^2 + \frac {1 \times 3 \times 5} {2 \times 4 \times 6} x^3 + \cdots\) | simplifying |
$\blacksquare$
Proof 2
| \(\ds \frac 1 {\sqrt {1 - x} }\) | \(=\) | \(\ds \paren {1 - x}^{-\frac 1 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {-\frac 1 2}^{\underline k} } {k!} \paren {-x}^k\) | General Binomial Theorem | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 0}^{k - 1} \paren {\paren {-\frac 1 2} - j} } {k!} \paren {-x}^k\) | Definition of Falling Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\ds \prod_{j \mathop = 1}^k \paren {2 j - 1} } {2^k k!} \paren {-x}^k\) | Translation of Index Variable of Product and simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \paren {-1}^k \frac {\ds \prod_{j \mathop = 1}^k \paren {2 j - 1} \prod_{j \mathop = 1}^k \paren {2 j} } {2^k k! \ds \prod_{j \mathop = 1}^k \paren {2 j} } \paren {-x}^k\) | multiplying top and bottom by $\ds \prod_{j \mathop = 1}^k \paren {2 j}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\ds \prod_{j \mathop = 1}^{2 k} j} {2^k k! \paren {2^k \ds \prod_{j \mathop = 1}^k j} } x^k\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {2 k}!} {2^k k! \paren {2^k k!} } x^k\) | Definition of Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {\paren {2 k}!} {\paren {2^k k!}^2} x^k\) | simplifying |
$\blacksquare$