To expand $(4 + x)^{-\frac{1}{2}}$ in ascending powers of $x$, we can use the binomial series expansion.

The general binomial expansion for $(a + b)^n$ is:

$(a + b)^n = a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots$

For $(4 + x)^{-\frac{1}{2}}$, we have $a = 4$, $b = x$, and $n = -\frac{1}{2}$.

Now, apply the binomial expansion up to the $x^4$ term.

Step-by-step Expansion:

Start with $a = 4$ and $n = -\frac{1}{2}$. We expand around $x = 0$.

$(4 + x)^{-\frac{1}{2}} = 4^{-\frac{1}{2}} \left(1 + \frac{x}{4}\right)^{-\frac{1}{2}}$ Since $4^{-\frac{1}{2}} = \frac{1}{2}$, the expression becomes:

$\frac{1}{2} \left(1 + \frac{x}{4}\right)^{-\frac{1}{2}}$

Now expand $\left(1 + \frac{x}{4}\right)^{-\frac{1}{2}}$ using the binomial expansion formula.

$(1 + u)^{-\frac{1}{2}} = 1 - \frac{1}{2}u + \frac{3}{8}u^2 - \frac{5}{16}u^3 + \frac{35}{128}u^4 + \cdots$

Substitute $u = \frac{x}{4}$ into the expansion:

$\left(1 + \frac{x}{4}\right)^{-\frac{1}{2}} = 1 - \frac{1}{2} \cdot \frac{x}{4} + \frac{3}{8} \cdot \left(\frac{x}{4}\right)^2 - \frac{5}{16} \cdot \left(\frac{x}{4}\right)^3 + \frac{35}{128} \cdot \left(\frac{x}{4}\right)^4 + \cdots$

Simplify each term:

$= 1 - \frac{x}{8} + \frac{3x^2}{128} - \frac{5x^3}{1024} + \frac{35x^4}{16384} + \cdots$

Multiply by $\frac{1}{2}$ to get the final expansion:

$(4 + x)^{-\frac{1}{2}} = \frac{1}{2} \left( 1 - \frac{x}{8} + \frac{3x^2}{128} - \frac{5x^3}{1024} + \frac{35x^4}{16384} \right)$

Now distribute $\frac{1}{2}$:

$(4 + x)^{-\frac{1}{2}} = \frac{1}{2} - \frac{x}{16} + \frac{3x^2}{256} - \frac{5x^3}{2048} + \frac{35x^4}{32768}$

Final Expansion (up to $x^4$):

$(4 + x)^{-\frac{1}{2}} = \frac{1}{2} - \frac{x}{16} + \frac{3x^2}{256} - \frac{5x^3}{2048} + \frac{35x^4}{32768}$

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Would you like a more detailed explanation of any specific step, or do you have any further questions about binomial expansions?

Here are some related questions to explore:

1. How do we calculate the binomial coefficients for negative exponents?
2. Can the binomial expansion be used for non-integer powers of a binomial expression?
3. How do you handle terms in a binomial series expansion when the exponent is a fraction?
4. What happens if we expand the same expression for higher powers of $x$?
5. How do you find the radius of convergence for a binomial series?

Tip: For a binomial expansion to be valid, the absolute value of $\frac{x}{4}$ must be less than 1, so the expansion is valid for $|x| < 4$.