General Binomial Expansion - Subtleties (Edexcel A-Level Mathematics): Flashcards

$|ax| < 1$ ensures the expansion converges and is valid.

$|x| < 1/8$ for the expansion of $\sqrt{1 + 8x}$ to hold.

$|x| < \frac{1}{7}$ makes the expansion of $\frac{4}{\sqrt[3]{1 - 7x}}$ valid.

Rewrite as $(1 + (2/3)x)^{-2}$ and then apply the binomial.

The first term is $1$ for the expansion of $(1 - 2x)^{-1/2}$.

The coefficient is $3/2$ after binomial series expansion.

$|x| < 1$ ensures convergence of the series.

$|ax|$ must be less than 1 for the expansion to be valid.

$|x| < 3/2$ keeps the expansion valid; powers of $x$ converge.

Factor out $3^{-2}$, then apply binomial expansion.