please help ---### Power Series Expansion with Center \( c = -7 \)To expand the function \( \frac{7}{1-x} \) in a power series with the center \( c = -7 \), use the equation:\[ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad \text{for} \quad |x| < 1 \]Since we want to expand \( \frac{7}{1-x} \), we realize that it can be expressed as \[ 7 \cdot \frac{1}{1-x} \]Therefore, we substitute \( 7 \cdot \sum_{n=0}^{\infty} x^n \) for \( \frac{7}{1-x} \):\[ \frac{7}{1-x} = 7 \sum_{n=0}^{\infty} x^n \]Given that the center of the power series is \( c = -7 \), which implies a shift in the variable \(x\):\[ x \rightarrow (x + 7) \]This leads to the expansion:\[ \frac{7}{1-(x + 7)} = 7 \sum_{n=0}^{\infty} (x + 7)^n\]However, the following expression shown is incorrect:\[ \frac{7}{1-x} = \sum_{n=0}^{\infty} 8^{1-n} (7 + x)^n \]The correct power series expansion should be:\[ \frac{7}{1-(x + 7)} = 7 \sum_{n=0}^{\infty} (x + 7)^n \]Keep in mind the conditions under which this series converges: \[ |x + 7| < 1 \]

Name: please help ---### Power Series Expansion with Center \( c = -7 \)To e
Uploaded: 2024-03-20T07:07:37.000Z
Channel: Bartleby
Description: please help ---### Power Series Expansion with Center \( c = -7 \)To expand the function \( \frac{7}{1-x} \) in a power series with the center \( c = -7 \), use the equation:\[ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad \text{for} \quad |x| < 1 \]