Find the first three terms, in ascending powers of x, of the binomial expansion of 1/(√4 + x).

Steps to Solve

1. Identify the expression to expand

We start with the expression ( \frac{1}{\sqrt{4} + x} ). We can rewrite this as ( (\sqrt{4} + x)^{-1} ).

2. Apply the Binomial Theorem

The Binomial Theorem states that for any real number ( n ) and ( |x| < 1 ): $$ (1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k $$

We will apply it here. Set ( a = \sqrt{4} ) and ( b = x ): $$ a + b = 2 + x $$

3. Rewrite the expression using the Binomial Theorem

We want to expand ( (2 + x)^{-1} ). To match the Binomial format, we can factor ( 2 ) out: $$ (2 + x)^{-1} = \frac{1}{2} \left( 1 + \frac{x}{2} \right)^{-1} $$

4. Use the Binomial expansion

Using the Binomial Theorem, we expand ( (1 + u)^{-1} ) where ( u = \frac{x}{2} ): $$ (1 + u)^{-1} = \sum_{k=0}^{\infty} (-1)^k u^k = 1 - u + u^2 - u^3 + \cdots $$

5. Find the first three terms

Substituting back ( u = \frac{x}{2} ):

• For ( k = 0 ): ( 1 )
• For ( k = 1 ): ( -\frac{x}{2} )
• For ( k = 2 ): ( \left(\frac{x}{2}\right)^2 = \frac{x^2}{4} )

Thus, the first three terms of the expansion are: $$ 1 - \frac{x}{2} + \frac{x^2}{4} $$

6. Multiply by the factor outside

Finally, since we factored out ( \frac{1}{2} ): $$ \frac{1}{2} (1 - \frac{x}{2} + \frac{x^2}{4}) = \frac{1}{2} - \frac{x}{4} + \frac{x^2}{8} $$