To find the power series representation for the function $f(x) = \frac{2}{1 + x}$ and determine its radius of convergence, we can proceed by expressing the function in a form that allows us to use a standard geometric series expansion.

Step 1: Rewrite the Function

The function can be rewritten as: $f(x) = \frac{2}{1 + x} = 2 \cdot \frac{1}{1 - (-x)}$ This is now in the form of a geometric series, $\frac{1}{1 - u} = \sum_{n=0}^{\infty} u^n$, where $|u| < 1$.

Step 2: Substitute $u = -x$

Here, $u = -x$, so we can express $f(x)$ as: $f(x) = 2 \sum_{n=0}^{\infty} (-x)^n$

Step 3: Expand the Series

Expanding this series, we get: $f(x) = 2 \sum_{n=0}^{\infty} (-1)^n x^n$ which simplifies to: $f(x) = \sum_{n=0}^{\infty} 2(-1)^n x^n$

Thus, the power series representation for $f(x) = \frac{2}{1 + x}$ is: $f(x) = \sum_{n=0}^{\infty} 2(-1)^n x^n$

Step 4: Determine the Radius of Convergence

Since this series is based on the geometric series $\frac{1}{1 - u}$ with $u = -x$, it converges when $|u| < 1$, or: $|x| < 1$

Therefore, the radius of convergence $R$ is: $R = 1$

Final Answer

The power series representation for $f(x) = \frac{2}{1 + x}$ is: $f(x) = \sum_{n=0}^{\infty} 2(-1)^n x^n$ with a radius of convergence $R = 1$.

Would you like more details on any of these steps or have any questions?

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1. What is the general form of a geometric series and how does it help in series expansion?
2. How do you find the radius of convergence for any power series?
3. What is the difference between convergence of a series and radius of convergence?
4. How does changing the function in the denominator affect the radius of convergence?
5. Can this series representation be used for functions other than rational functions?

Tip: Always check the form of the function against known series expansions (like geometric series) to simplify the process of finding power series representations.