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)−n=1∑∞(xxxx)
Determine the radius of convergence, R.
R=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(1−11x)ln(1−t)=−n=1∑∞ntnfor∣t∣<1ln(1−11x)=−n=1∑∞n(11x)nf(x)=ln(11)−n=1∑∞11nnxn The radius of convergence R is determined by the condition 11x<1: 11x<1⟹∣x∣<11R=11Final Answers: 1. The power series representation for f(x)=ln(11−x) is: f(x)=ln(11)−n=1∑∞11nnxn 2. The radius of convergence R is: R=11