Given the function: $$f(x) = \frac{1}{\sqrt{9 + 4x^2}}, \quad |x| < \frac{3}{2}$$ Find the first three non-zero terms of the binomial expansion of $f(x)$ in ascending powers of $x$ - Edexcel - A-Level Maths Pure - Question 4 - 2011 - Paper 5

Using the binomial expansion for $(1 + k)^n$ where $n = -\frac{1}{2}$ and $k = \frac{4}{9}x^2$, we have:

$(1 + k)^n = 1 + nk + \frac{n(n - 1)}{2}k^2 + \cdots$

For our case:

• The first term is $1$.
• The second term is $n \cdot k = -\frac{1}{2} \cdot \frac{4}{9}x^2 = -\frac{2}{9}x^2$
• The third term is $\frac{n(n - 1)}{2} \cdot k^2 = \frac{-\frac{1}{2}(-\frac{3}{2})}{2} \cdot \left(\frac{4}{9} x^2\right)^2 = \frac{3}{2} \cdot \frac{16}{81} x^4 = \frac{24}{162} x^4 = \frac{4}{27} x^4$

Now we can summarize the first three non-zero terms:

$f(x) = \frac{1}{3} \left(1 - \frac{2}{9}x^2 + \frac{4}{27}x^4 + \cdots\right)$

Therefore, the first three non-zero terms are:

• The coefficient of $x^0$ is rac{1}{3}
• The coefficient of $x^2$ is $-\frac{2}{27}$
• The coefficient of $x^4$ is rac{4}{81}