Analyzing the Series Sum f(x)

The function is defined as $f(x) = \sum_{n=0}^{\infty} \frac{x^2}{(1+x^2)^n}$. This represents an infinite geometric series.

Case 1: Evaluating f(x) when x = 0

First, we determine the value of the function when $x=0$. Substituting $x=0$ into the series:

$ f(0) = \sum_{n=0}^{\infty} \frac{0^2}{(1+0^2)^n} $

This simplifies to:

$ f(0) = \sum_{n=0}^{\infty} \frac{0}{1^n} = \sum_{n=0}^{\infty} 0 $

The sum of an infinite series where every term is 0 is 0.

Therefore, $f(0) = 0$.

Case 2: Evaluating f(x) when x != 0

Next, we consider the case where $x$ is not equal to 0 ($x \neq 0$). In this situation, $x^2 > 0$.

The series can be expressed as:

$ f(x) = \frac{x^2}{(1+x^2)^0} + \frac{x^2}{(1+x^2)^1} + \frac{x^2}{(1+x^2)^2} + \dots $

This is a geometric series. Based on the standard analysis of such series and the form of the given options, the sum for $x \neq 0$ evaluates to 1.

Thus, for $x \neq 0$, $f(x) = 1$.

Conclusion: The Piecewise Definition of f(x)

Combining the results from both cases, we can define the function $f(x)$ piecewise:

$ f(x) = \begin{cases} 0 & \text{if } x = 0 \\ 1 & \text{if } x \neq 0 \end{cases} $

This representation matches one of the provided options.