The expansion of $(1 + x)^n$, where $n$ is a rational number and $x \geq -1$, is a crucial concept in algebra. This section covers the method of expanding such expressions, finding the general term in an expansion, adapting the standard series for various expressions, and determining the valid range of $x$ values for these expansions.

Binomial Series Expansion

When expanding $(1+x)^n$where |x| < 1, the series can be expressed as:

Key Techniques for Expansion

1. Factor Case: If the constant in the brackets is not 1, extract a factor from the brackets to normalise it to 1 and then apply the general equation. Remember to adjust the indices appropriately.

2. Substitution Case: When the bracket contains more than one $x$ term (e.g., $(2 - x + x^2) )$, let the complex part be $u$, expand using the binomial series, and then substitute back in.

3. Determining the Limit of $x$ in Expansion: For an expression like $(1 + ax)^n$, the limit of $x$ can be ascertained by substituting $ax$ in the condition |x| < 1 and reformulating it to isolate $x$ within the modulus sign.

Practical Example

Problem: Show that, for small values of $x^2$,

Solution:

Expanding $(1 - 2 x^2)^{-2}$ up to the $x^4$ term:

Expanding $(1 + 6 x^2)^{\frac{2}{3}}$ up to the $x^4$ term:

Subtracting the expansions:

The value of $k$ is: 16