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To find the power series representation of the function \( \frac{1}{1 + 5x} \) centered at \( x = 2 \), we can follow these steps:

1. **Rewrite the Function for Series Expansion:**
   A standard power series expansion around \( x = 0 \) for \( \frac{1}{1 - u} \) is given by:
   \[
   \frac{1}{1 - u} = \sum_{n=0}^{\infty} u^n, \quad \text{for } |u| < 1.
   \]

2. **Express in Terms of a New Variable:**
   We want a series centered at \( x = 2 \). Define a new variable \( u \) such that:
   \[
   u = -5(x - 2).
   \]
   Then:
   \[
   1 + 5x = 1 - (-5(x - 2)) = 1 - u.
   \]
   Now our function becomes:
   \[
   \frac{1}{1 + 5x} = \frac{1}{1 - u} = \sum_{n=0}^{\infty} (-5(x - 2))^n.
   \]

3. **Expand the Series:**
   Substituting \( u = -5(x - 2) \), we get:
   \[
   \sum_{n=0}^{\infty} (-5)^n (x - 2)^n.
   \]

So, the power series representation centered at \( x = 2 \) for \( \frac{1}{1 + 5x} \) is:
\[
\sum_{n=0}^{\infty} (-5)^n (x - 2)^n.
\]

Would you like further clarification on any steps?

### Related Questions
1. How do you find the radius of convergence for this power series?
2. How does the power series change if the center is at a different point?
3. How can the ratio test be applied to confirm convergence of this series?
4. What is the general formula for a Taylor series expansion of a function around a point?
5. How does the presence of \(5x\) affect the convergence properties of this series?

### Tip
Remember, a power series is valid within the interval where the series converges; finding this interval is key for accurate results.

Which of the following is the power series representation centered at x = 2 for the function: 1 / (1 + 5x)?

Learn how to find the power series representation of the function 1 / (1 + 5x) centered at x = 2. This solution includes series expansion techniques and the use of Taylor series, focusing on algebraic manipulation with convergence conditions. Suitable for high school students in Grades 11-12.

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