General Binomial Theorem/Examples/(1+5x)^(1/5)

Example of Use of General Binomial Theorem

$\paren {1 + 5 x}^{\frac 1 5} = 1 + x - 2 x^2 + 6 x^3 + \cdots$

Proof

$\ds \paren {1 + 5 x}^{\frac 1 5}$

$=$

$\ds \sum_{n \mathop = 0}^\infty \frac {\paren {\frac 1 5}^{\underline n} } {n!} \paren {5 x}^n$

General Binomial Theorem

$\ds $

$=$

$\ds 1 + \paren {\frac 1 5} \paren {5 x} + \dfrac {\paren {\frac 1 5} \paren {-\frac 4 5} } {2!} \paren {5 x}^2 + \dfrac {\paren {\frac 1 5} \paren {-\frac 4 5} \paren {-\frac 9 5} } {3!} \paren {5 x}^3 + \cdots$

expanding term by term

$\ds $

$=$

$\ds 1 + x + \dfrac {\paren {-4} } {5^2 \times 2!} \paren {5 x}^2 + \dfrac {\paren {-4} \paren {-9} } {5^3 \times 3!} \paren {5 x}^3 + \cdots$

simplifying

$\ds $

$=$

$\ds 1 + x - 2 x^2 + 6 x^3 + \cdots$

simplifying

$\blacksquare$

Sources

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