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.