**f(x)=**

1−

x

2/

1+

x

2




.

Question 1

Select one:




a.

**−1+2**

**∑**

∞

n=0

(−1

)

n

x

2n




b.

**−1+2**

**∑**

∞

n=0

x

n




c.

**−1+2**

**∑**

∞

n=0

(−1

)

n

x

n




d.

**−1+2**

**∑**

∞

n=0

x

2n

To solve this problem, we need to find the correct series expansion for the function:

\[
f(x) = \frac{1 - x^2}{1 + x^2}.
\]

### Step-by-step Solution:

1. **Rewrite the function:**

   Notice that the function can be rewritten in a more manageable form:
   \[
   f(x) = \frac{1 - x^2}{1 + x^2} = \frac{1 + x^2 - 2x^2}{1 + x^2} = 1 - \frac{2x^2}{1 + x^2}.
   \]

   So, we can express \(f(x)\) as:
   \[
   f(x) = 1 - 2 \cdot \frac{x^2}{1 + x^2}.
   \]

2. **Find the series expansion for \(\frac{1}{1 + x^2}\):**

   We recognize that \(\frac{1}{1 + x^2}\) is a geometric series for \(|x^2| < 1\):
   \[
   \frac{1}{1 + x^2} = \sum_{n=0}^{\infty} (-1)^n x^{2n}.
   \]

3. **Substitute the series into \(f(x)\):**

   Now, substitute the series expansion into the expression for \(f(x)\):

   \[
   f(x) = 1 - 2 \cdot \left(x^2 \sum_{n=0}^{\infty} (-1)^n x^{2n}\right).
   \]

   This simplifies to:

   \[
   f(x) = 1 - 2 \sum_{n=0}^{\infty} (-1)^n x^{2n+2}.
   \]

4. **Adjust the index for the series:**

   To match the series to the options given, we change the index in the series by substituting \(m = n+1\):

   \[
   f(x) = 1 - 2 \sum_{m=1}^{\infty} (-1)^{m-1} x^{2m}.
   \]

   Simplify the exponent signs:

   \[
   f(x) = 1 - 2 \sum_{n=0}^{\infty} (-1)^n x^{2n}.
   \]

5. **Combine the constant term:**

   The series starts from \(n = 0\), so it becomes:

   \[
   f(x) = -1 + 2 \sum_{n=0}^{\infty} (-1)^n x^{2n}.
   \]

### Conclusion:

This matches with **Option A**:

\[
\textbf{A. } -1 + 2 \sum_{n=0}^{\infty} (-1)^n x^{2n}.
\]

### Final Answer:

**The correct answer is (a).**

Would you like further details or explanations? Here are 5 related questions you might find interesting:

1. How do you derive the geometric series formula?
2. What are the convergence conditions for a series expansion?
3. How do you change the index of summation in a series?
4. Can you derive Taylor series for more complex functions?
5. What is the radius of convergence for a series like \(\sum (-1)^n x^{2n}\)?

**Tip:** When working with series, always check the radius of convergence to ensure the series representation is valid.

Explore the series expansion for the function f(x) = (1 - x^2) / (1 + x^2). Learn how to derive the series using the geometric series formula and understand its convergence conditions. Suitable for undergraduate level studies.

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