2 Zigmas Apart

Calculus Level 5

\[\displaystyle \sum_{n=0}^{\infty} \dfrac{(2n-1)!!}{(2n)!!} = 1 + \dfrac{1}{2}+ \dfrac{1\cdot 3}{2\cdot 4} + \dfrac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6} + \cdots\]

\[\displaystyle \sum_{n=0}^{\infty} x^n = 2 + \displaystyle \sum_{n=0}^{\infty} \dfrac{(2n-1)!!}{(2n)!!}x^n\]

Find the positive value of \(x\) satisfying the equation above.

Notation: \(!!\) denotes the double factorial notation. For example, \(10!!=10\times8\times6\times4\times2 \).

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