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 centered at \( x = 0 \).)

\[ f(x) = \ln(11 - x) \]

\[ f(x) = \ln(11) - \sum_{n=1}^{\infty} \left( \boxed{\phantom{xxxx}} \right) \]

Determine the radius of convergence, \( R \).

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

Solution

To find the power series representation for \( f(x) = \ln(11 - x) \) and determine its radius of convergence, follow these steps: \[ f(x) = \ln(11) + \ln\left(1 - \frac{x}{11}\right) \] \[ \ln(1 - t) = -\sum_{n=1}^{\infty} \frac{t^n}{n} \quad \text{for} \ |t| < 1 \] \[ \ln\left(1 - \frac{x}{11}\right) = -\sum_{n=1}^{\infty} \frac{\left(\frac{x}{11}\right)^n}{n} \] \[ f(x) = \ln(11) - \sum_{n=1}^{\infty} \frac{x^n}{11^n n} \] The radius of convergence \( R \) is determined by the condition \( \left|\frac{x}{11}\right| < 1 \): \[ \left|\frac{x}{11}\right| < 1 \implies |x| < 11 \] \[ R = 11 \] Final Answers: 1. The power series representation for \( f(x) = \ln(11 - x) \) is: \[ f(x) = \ln(11) - \sum_{n=1}^{\infty} \frac{x^n}{11^n n} \] 2. The radius of convergence \( R \) is: \[ R = 11 \]