To solve this problem, we need to find the value of $a$ such that the coefficient of $x^3$ in the series expansion of

$f(x) = \frac{(4 - x)(a - x)}{(1 + x^2)^{1/3}}$

is zero. Here's how we proceed:

Step 1: Expand the numerator

The numerator is $(4 - x)(a - x)$. Expanding this:

$(4 - x)(a - x) = 4a - (4 + a)x + x^2$

So the numerator becomes $4a - (4 + a)x + x^2$.

Step 2: Expand the denominator

The denominator is $(1 + x^2)^{1/3}$. Using a binomial series expansion for $(1 + x^2)^{1/3}$ (valid for small $x^2$):

$(1 + x^2)^{1/3} \approx 1 + \frac{1}{3}x^2 - \frac{1}{9}x^4 + \cdots$

Step 3: Write $f(x)$ as a series

Substitute the expanded numerator and denominator into $f(x)$:

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

We perform the division by expanding in powers of $x$:

1. Divide the numerator by $1$: $4a - (4 + a)x + x^2$.
2. Subtract higher-order terms from the series expansion.

After matching terms, collect the coefficient of $x^3$.

Step 4: Coefficient of $x^3$

The coefficient of $x^3$ will involve contributions from terms like:

• The $x^2$ in the numerator times the $\frac{1}{3}x^2$ from the denominator expansion.

You can simplify this by collecting all terms contributing to $x^3$. Setting the coefficient of $x^3$ to zero allows you to solve for $a$.

Let me calculate this explicitly.from sympy import symbols, series, expand, collect

Define variables

x, a = symbols('x a')

Define the numerator and denominator

numerator = (4 - x) * (a - x) denominator = (1 + x2)(1/3)

Define the function f(x)

f_x = numerator / denominator

Expand f(x) as a series up to x^4

f_x_series = series(f_x, x, n=4)

Collect the coefficient of x^3

f_x_series_expanded = expand(f_x_series) coeff_x3 = collect(f_x_series_expanded, x).coeff(x, 3)

coeff_x30.333333333333333a + 1.333333333333330.333333333333333a + 1.33333333333333successThe coefficient of $x^3$ in the series expansion is:

$\frac{1}{3}a + \frac{4}{3}$

Step 5: Set the coefficient to zero

To find $a$, set this coefficient to zero:

$\frac{1}{3}a + \frac{4}{3} = 0$

Solving this equation:

$a = -4$

Thus, the value of the constant $a$ is $a = -4$.

Would you like further details or explanations? Here are some related questions for deeper understanding:

1. How is the binomial expansion for $(1 + x^2)^{1/3}$ derived?
2. What are the conditions for the validity of a series expansion like this?
3. Can this approach be used for non-integer powers of the denominator?
4. How do the coefficients of other terms (e.g., $x^2$, $x^4$) contribute to the expansion?
5. What happens if the denominator has a different exponent, e.g., $1/2$?

Tip: Always verify the series expansion for both numerator and denominator to ensure higher-order terms don't interfere with lower-order terms' coefficients.