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Let's work on this problem step by step.

You are given the function:

\[
f(x) = \frac{1}{5 + x}
\]

### (a) Geometric Power Series Expansion:
The goal is to express \( f(x) \) as a power series centered at 0. First, we need to rewrite the function to resemble the standard geometric series form:

\[
\frac{1}{1 - r} = \sum_{n=0}^{\infty} r^n \quad \text{for} \ |r| < 1.
\]

We start by manipulating the function:

\[
f(x) = \frac{1}{5 + x} = \frac{1}{5(1 + \frac{x}{5})}.
\]

Now we can factor out \( \frac{1}{5} \):

\[
f(x) = \frac{1}{5} \cdot \frac{1}{1 - (-\frac{x}{5})}.
\]

This fits the geometric series form \( \frac{1}{1 - r} \) with \( r = -\frac{x}{5} \). Using the geometric series formula, we can expand \( f(x) \) as:

\[
f(x) = \frac{1}{5} \sum_{n=0}^{\infty} \left(-\frac{x}{5}\right)^n = \sum_{n=0}^{\infty} \frac{(-1)^n}{5^{n+1}} x^n.
\]

This is the required geometric power series expansion.

### (b) Long Division Method:
For long division, we divide \( 1 \) by \( 5 + x \). Let's calculate the first three terms.

\[
\frac{1}{5 + x} = \frac{1}{5} \cdot \frac{1}{1 + \frac{x}{5}}.
\]

Perform long division of \( 1 \) by \( 5 + x \) to obtain the first three terms. The division yields:

\[
\frac{1}{5 + x} = \frac{1}{5} - \frac{x}{25} + \frac{x^2}{125} + \cdots.
\]

Thus, the first three terms of the series are:

\[
f(x) = \frac{1}{5} - \frac{x}{25} + \frac{x^2}{125}.
\]

Would you like further details or have any questions about this process?

### Relative Questions:
1. How does the geometric series formula apply to rational functions like \( f(x) = \frac{1}{5 + x} \)?
2. What are the conditions under which the geometric series converges?
3. How would you calculate more terms using the long division method?
4. How can you verify the convergence of the geometric series derived in part (a)?
5. What changes in the series expansion if the function is centered at a different point, say \( x = 1 \)?

### Tip:
When working with power series, always check the radius of convergence to ensure the series is valid in the region you are working with.

Find a geometric power series for the function f(x) = 1 / (5 + x), centered at 0, using the following methods: (a) by the technique shown in Examples 1 and 2, and (b) by long division (give the first three terms).

Learn how to find the geometric power series for the function f(x) = 1 / (5 + x), centered at 0, through two methods: geometric series formula and long division. This detailed solution covers essential algebraic concepts and is suitable for advanced high school or early college students.

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