Question: Find a power series representation for the function. (Give your power series representation.) \[…

Find a power series representation for the function. (Give your power series representation.)

\[ f(x) = \frac{4}{5 - x} \]

\[ f(x) = \sum_{n=0}^{\infty} \left( \boxed{\phantom{xxxx}} \right) \]

Determine the interval of convergence. (Enter your answer using interval notation.)

\[ \boxed{\phantom{xxxx}} \]

Solution

Rewrite the function to match the geometric series form: \[ f(x) = \frac{4}{5 - x} = \frac{4}{5} \cdot \frac{1}{1 - \left( \frac{x}{5} \right)} \] Use the geometric series expansion: \[ \frac{1}{1 - r} = \sum_{n=0}^{\infty} r^n & \] Substitute \( r = \frac{x}{5} \): \[ f(x) = \frac{4}{5} \sum_{n=0}^{\infty} \left( \frac{x}{5} \right)^n & \] Determine the interval of convergence: \[ \left| \frac{x}{5} \right| < 1 \implies |x| < 5 & \] Power series representation: \[ f(x) = \frac{4}{5} \sum_{n=0}^{\infty} \left( \frac{x}{5} \right)^n \] Interval of convergence: \[ |x| < 5 \]