find a power series centered at 0 for function: f(x)= 1/(1+13x)^2

To find the power series centered at 0 for the function \( f(x) = \frac{1}{(1+13x)^2} \), we can proceed by following these steps:

1. **Identify the Known Series Expansion**: Start with the geometric series:
   \[
   \frac{1}{1 - u} = \sum_{n=0}^{\infty} u^n \quad \text{for } |u| < 1.
   \]

2. **Rewrite the Function**: Rewrite \( f(x) \) in a form that resembles the geometric series. Let \( u = -13x \), so that:
   \[
   \frac{1}{1 + 13x} = \frac{1}{1 - (-13x)} = \sum_{n=0}^{\infty} (-13x)^n = \sum_{n=0}^{\infty} (-13)^n x^n.
   \]

3. **Differentiate to Get \( \frac{1}{(1+13x)^2} \)**: Since our target function is \( \frac{1}{(1+13x)^2} \), we differentiate the geometric series term by term with respect to \( x \):
   \[
   \frac{d}{dx} \left( \frac{1}{1 + 13x} \right) = \frac{d}{dx} \left( \sum_{n=0}^{\infty} (-13)^n x^n \right).
   \]

   Differentiating the left side gives:
   \[
   \frac{d}{dx} \left( \frac{1}{1 + 13x} \right) = -\frac{13}{(1 + 13x)^2}.
   \]
   
   To match \( f(x) = \frac{1}{(1 + 13x)^2} \), divide by \(-13\):
   \[
   \frac{1}{(1 + 13x)^2} = \sum_{n=1}^{\infty} n(-13)^n x^{n-1}.
   \]

So the power series for \( f(x) = \frac{1}{(1+13x)^2} \) is:
\[
f(x) = \sum_{n=1}^{\infty} n(-13)^n x^{n-1}.
\]

### Do you want more details on these steps or have questions? 

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#### 5 Related Questions:

1. How would the series change if the function were \( \frac{1}{(1+ax)^2} \) with a different constant \( a \)?
2. How can we find the radius of convergence for this series?
3. What happens if we want a power series for \( \frac{1}{(1+13x)^3} \) instead?
4. How does differentiation affect the convergence of a series?
5. Could we obtain this series by expanding \( \frac{1}{1+13x} \) and then squaring it?

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#### One Tip:
To derive power series, rewriting the function in terms of simpler series forms, like the geometric series, often simplifies differentiation and integration.

Find a power series centered at 0 for function: f(x) = 1/(1+13x)^2

Explore how to find the power series centered at 0 for the function f(x) = 1/(1+13x)^2. This solution includes the application of geometric series expansion, differentiation term by term, and provides a step-by-step explanation suitable for college-level students.

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