Thanks for pointing that out. Let's solve the problem with the corrected function: \(\frac{2x - 6}{x^2 - 6x + 5}\).

### Step 1: Partial Fraction Decomposition

First, factor the denominator:

\[ 
x^2 - 6x + 5 = (x - 1)(x - 5).
\]

Thus, we write:

\[
\frac{2x - 6}{(x - 1)(x - 5)} = \frac{A}{x - 1} + \frac{B}{x - 5}.
\]

To find \(A\) and \(B\), set up the equation:

\[
2x - 6 = A(x - 5) + B(x - 1).
\]

Expanding and simplifying:

\[
2x - 6 = Ax - 5A + Bx - B = (A + B)x - (5A + B).
\]

Equating coefficients, we get:

1. \(A + B = 2\)
2. \(-5A - B = -6\)

Solve these equations simultaneously:

1. From \(A + B = 2\), we have \(B = 2 - A\).
2. Substitute into the second equation: \(-5A - (2 - A) = -6\).

Simplify:

\[
-5A - 2 + A = -6 \quad \Rightarrow \quad -4A = -4 \quad \Rightarrow \quad A = 1.
\]

Then, \(B = 2 - 1 = 1\).

So, the partial fraction decomposition is:

\[
\frac{2x - 6}{(x - 1)(x - 5)} = \frac{1}{x - 1} + \frac{1}{x - 5}.
\]

### Step 2: Express as Power Series

Now, express each term as a power series centered at \(x = 0\):

1. For \(\frac{1}{x - 1}\), rewrite it as \(-\frac{1}{1 - x}\). The geometric series expansion is:

   \[
   \sum_{n=0}^{\infty} (-1)x^n = -\sum_{n=0}^{\infty} x^n \quad \text{for} \quad |x| < 1.
   \]

2. For \(\frac{1}{x - 5}\), rewrite it as \(-\frac{1}{5} \cdot \frac{1}{1 - \frac{x}{5}}\). The geometric series expansion is:

   \[
   \sum_{n=0}^{\infty} \left(-\frac{1}{5}\right) \left(\frac{x}{5}\right)^n = -\sum_{n=0}^{\infty} \frac{x^n}{5^{n+1}} \quad \text{for} \quad |x/5| < 1 \Rightarrow |x| < 5.
   \]

### Step 3: Power Series Representation

Combining both series:

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

Simplifying the series:

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

### Step 4: Interval of Convergence

The interval of convergence is determined by the stricter condition of the two series:

1. \(|x| < 1\) for \(\frac{1}{x - 1}\).
2. \(|x| < 5\) for \(\frac{1}{x - 5}\).

Therefore, the interval of convergence for the combined series is \(|x| < 1\).

### Final Answer

- **Power Series Representation:**
  \[
  E(x) = -\sum_{n=0}^{\infty} \left(1 + \frac{1}{5^{n+1}}\right)x^n
  \]

- **Interval of Convergence:** \((-1, 1)\)

If you have any questions or need further details, feel free to ask!

Here are some additional questions to explore:

1. How do partial fractions help simplify rational functions?
2. What are the limitations of using partial fractions for series expansion?
3. How do you determine the convergence of a power series?
4. What other methods can you use to find the interval of convergence?
5. Can you apply partial fraction decomposition to non-polynomial functions?
6. How does changing the center of a power series affect convergence?
7. What is the role of the geometric series in calculus?
8. How does the radius of convergence relate to complex analysis?

**Tip:** Always verify the interval of convergence by checking the series behavior at the boundaries, if necessary!

Learn how to solve the rational function 2x - 6 / x^2 - 6x + 5 using partial fraction decomposition and power series expansion. This detailed guide covers the steps involved, including factorization, series representation, and convergence analysis.

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