f(x)=\dfrac{1}{1+6x}**Find a power series for \[f\].**

Choose 1 answer:

Choose 1 answer:

(Choice A)   

\[1+6x+36x^2+\ldots +6^n x ^n+\ldots\]

A

\[1+6x+36x^2+\ldots +6^n x ^n+\ldots\]

(Choice B)   

\[1+6x-36x^2+\ldots +(-6)^n x ^n+\ldots\]

B

\[1+6x-36x^2+\ldots +(-6)^n x ^n+\ldots\]

(Choice C)   

\[1-6x-36x^2+\ldots -6^n x ^{n}+\ldots\]

C

\[1-6x-36x^2+\ldots -6^n x ^{n}+\ldots\]

(Choice D)   

\[1-6x+36x^2+\ldots +(-6)^n x ^{n}+\ldots\]

D

\[1-6x+36x^2+\ldots +(-6)^n x ^{n}+\ldots\]

To find a power series for the function \( f(x) = \frac{1}{1 + 6x} \), we can start by recognizing that this function is similar to the standard geometric series formula:

\[
\frac{1}{1 - r} = 1 + r + r^2 + r^3 + \ldots 
\]

This series converges when \( |r| < 1 \).

### Rewriting \( f(x) \)

We can rewrite \( f(x) = \frac{1}{1 + 6x} \) in a form that fits the geometric series formula by factoring out the negative sign in the denominator:

\[
f(x) = \frac{1}{1 - (-6x)}
\]

Here, the common ratio \( r \) is \( -6x \). Therefore, the power series expansion for \( f(x) \) is:

\[
f(x) = 1 + (-6x) + (-6x)^2 + (-6x)^3 + \ldots
\]

Simplifying each term, we get:

\[
f(x) = 1 - 6x + 36x^2 - 216x^3 + \ldots + (-6)^n x^n + \ldots
\]

### Matching with the given choices:

This matches with **Choice D**:

\[
\boxed{1 - 6x + 36x^2 + \ldots + (-6)^n x^n + \ldots}
\]

This is the correct power series representation for \( f(x) = \frac{1}{1 + 6x} \).

Would you like more details or have any questions? Here are some related questions to explore:

1. What is the interval of convergence for the power series of \( f(x) = \frac{1}{1 + 6x} \)?
2. How do you find the power series expansion for more complex functions?
3. What happens to the series when \( |6x| \geq 1 \)?
4. How would the power series change if the function were \( f(x) = \frac{1}{1 - 6x} \)?
5. Can you derive the power series for \( f(x) = \frac{1}{1 + ax} \) for any constant \( a \)?

**Tip:** Power series are particularly useful for approximating functions near a specific point, often near zero, where they converge.

Discover how to find the power series expansion for the function f(x) = 1 / (1 + 6x). Explore the step-by-step solution using the geometric series formula and understand the convergence conditions. Suitable for advanced mathematics students and enthusiasts.

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