To solve this problem, we will find the power series representation for the given function

$f(x) = \frac{x}{(1 + 4x)^2},$

and determine the radius of convergence. Here's the solution:

Step 1: Rewrite the function for a power series expansion

We start with:

$f(x) = \frac{x}{(1 + 4x)^2}.$

The term $(1 + 4x)^2$ can be expressed using the derivative of the geometric series. Recall the geometric series expansion for $\frac{1}{1 - u}$ is:

$\frac{1}{1 - u} = \sum_{n=0}^\infty u^n, \quad \text{for } |u| < 1.$

Substitute $u = -4x$, so:

$\frac{1}{1 + 4x} = \sum_{n=0}^\infty (-4x)^n = \sum_{n=0}^\infty (-4)^n x^n, \quad \text{for } |x| < \frac{1}{4}.$

Differentiating with respect to $x$, we get:

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

On the other hand, differentiating the series term-by-term gives:

$-\frac{1}{(1 + 4x)^2} = \frac{d}{dx} \left( \sum_{n=0}^\infty (-4)^n x^n \right) = \sum_{n=1}^\infty n (-4)^n x^{n-1}.$

Therefore:

$\frac{1}{(1 + 4x)^2} = \sum_{n=1}^\infty n (-4)^n x^{n-1}.$

Step 2: Multiply by $x$ to obtain $f(x)$

Now, multiply $x$ by the series for $\frac{1}{(1 + 4x)^2}$:

$f(x) = x \cdot \frac{1}{(1 + 4x)^2} = x \cdot \sum_{n=1}^\infty n (-4)^n x^{n-1}.$

Simplify the powers of $x$:

$f(x) = \sum_{n=1}^\infty n (-4)^n x^n.$

Step 3: Write the power series representation

Thus, the power series representation is:

$f(x) = \sum_{n=1}^\infty n (-4)^n x^n.$

Step 4: Determine the radius of convergence

The radius of convergence is determined by the term $(-4x)^n$. The geometric series $\sum_{n=0}^\infty (-4x)^n$ converges when:

$|4x| < 1 \implies |x| < \frac{1}{4}.$

Hence, the radius of convergence is:

$R = \frac{1}{4}.$

Final Answer:

1. Power Series Representation: $f(x) = \sum_{n=1}^\infty n (-4)^n x^n.$

2. Radius of Convergence: $R = \frac{1}{4}.$

Let me know if you'd like additional steps explained! Here are some related questions to explore:

1. How can we verify the convergence of this power series using the ratio test?
2. What happens to the power series if $f(x)$ is modified to $\frac{x^2}{(1 + 4x)^2}$?
3. How does differentiating or integrating this power series affect its radius of convergence?
4. What is the interval of convergence for this power series?
5. How can we use partial sums to approximate $f(x)$ for a specific value of $x$?

Tip: Always double-check the radius of convergence using the ratio test if you're unsure