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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 x = 0 .)

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

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

Determine the radius of convergence, R R .

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

Solution

To find the power series representation for f(x)=ln(11x) f(x) = \ln(11 - x) and determine its radius of convergence, follow these steps: f(x)=ln(11)+ln(1x11) f(x) = \ln(11) + \ln\left(1 - \frac{x}{11}\right) ln(1t)=n=1tnnfor t<1 \ln(1 - t) = -\sum_{n=1}^{\infty} \frac{t^n}{n} \quad \text{for} \ |t| < 1 ln(1x11)=n=1(x11)nn \ln\left(1 - \frac{x}{11}\right) = -\sum_{n=1}^{\infty} \frac{\left(\frac{x}{11}\right)^n}{n} f(x)=ln(11)n=1xn11nn f(x) = \ln(11) - \sum_{n=1}^{\infty} \frac{x^n}{11^n n} The radius of convergence R R is determined by the condition x11<1 \left|\frac{x}{11}\right| < 1 : x11<1    x<11 \left|\frac{x}{11}\right| < 1 \implies |x| < 11 R=11 R = 11 Final Answers: 1. The power series representation for f(x)=ln(11x) f(x) = \ln(11 - x) is: f(x)=ln(11)n=1xn11nn f(x) = \ln(11) - \sum_{n=1}^{\infty} \frac{x^n}{11^n n} 2. The radius of convergence R R is: R=11 R = 11

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