General Binomial Theorem/Examples/(1-4x)^(1/2)

Example of Use of General Binomial Theorem

$\paren {1 - 4 x}^{\frac 1 2} = 1 - 2 x - 2 x^2 + 4 x^3 + \cdots$

Proof

\(\ds \paren {1 - 4 x}^{\frac 1 2}\)

\(=\)

\(\ds \sum_{n \mathop = 0}^\infty \frac {\paren {\frac 1 2}^{\underline n} } {n!} \paren {-4 x}^n\)

General Binomial Theorem

\(\ds \)

\(=\)

\(\ds 1 + \paren {\frac 1 2} \paren {-4 x} + \dfrac {\paren {\frac 1 2} \paren {-\frac 1 2} } {2!} \paren {-4 x}^2 + \dfrac {\paren {\frac 1 2} \paren {-\frac 1 2} \paren {-\frac 3 2} } {3!} \paren {-4 x}^3 + \cdots\)

expanding term by term

\(\ds \)

\(=\)

\(\ds 1 - 2 x + \dfrac {\paren {-1} } {2^2 \times 2!} \paren {4 x}^2 + \dfrac 3 {2^3 \times 3!} \paren {4 x}^3 + \cdots\)

simplifying

\(\ds \)

\(=\)

\(\ds 1 - 2 x - 2 x^2 + 4 x^3 + \cdots\)

simplifying

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: The Binomial Theorem: Exercises $\text {III}$: $1 \ \text {(d)}$