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1. -/ 2.77 points

find a power series representation for the function. (give your power series representation centered at ( x = 0 ).)
( f(x)=\frac{6}{7 - x} )
( f(x)=sum_{n = 0}^{infty}(square) )
determine the interval of convergence. (enter your answer using interval notation.)

Explanation:

Step1: Rewrite the function

We know the geometric series formula \(\sum_{n = 0}^{\infty}r^{n}=\frac{1}{1 - r}\), for \(|r|\lt1\).
Rewrite \(f(x)=\frac{6}{7 - x}\) as \(f(x)=\frac{6}{7}\cdot\frac{1}{1-\frac{x}{7}}\).

Step2: Apply the geometric series formula

Let \(r = \frac{x}{7}\), then \(\frac{1}{1-\frac{x}{7}}=\sum_{n = 0}^{\infty}(\frac{x}{7})^{n}\), for \(|\frac{x}{7}|\lt1\).
So \(f(x)=\frac{6}{7}\sum_{n = 0}^{\infty}(\frac{x}{7})^{n}=\sum_{n = 0}^{\infty}\frac{6}{7^{n + 1}}x^{n}\).

Step3: Find the interval of convergence

We have \(|\frac{x}{7}|\lt1\).
Solve the inequality \(|\frac{x}{7}|\lt1\):
\(- 1\lt\frac{x}{7}\lt1\).
Multiply each part by \(7\): \(-7\lt x\lt7\).

Answer:

\(f(x)=\sum_{n = 0}^{\infty}\frac{6}{7^{n+1}}x^{n}\), interval of convergence is \((-7,7)\)