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Prove that the coefficient of x^r in the...

Prove that the coefficient of `x^r` in the expansion of `(1-2x)^(1//2)i s(2r)!//[2^r(r !)^2]dot`

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To prove that the coefficient of \( x^r \) in the expansion of \( (1 - 2x)^{\frac{1}{2}} \) is given by \( \frac{(2r)!}{2^r (r!)^2} \), we will use the Binomial Theorem for fractional powers. ### Step-by-Step Solution: 1. **Understanding the Binomial Expansion**: The Binomial Theorem states that for any real number \( n \) and \( |x| < 1 \): \[ (1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k ...
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