Activity $8\mathrm{.}4\mathrm{.}2.$ Consider the infinite Taylor series given by

$T(x)={\displaystyle \underset{k=1}{\overset{}{}}}\frac{1}{k*2^k}(x-1{)}^k$

$=\frac{1}{1*2}(x-1)+\frac{1}{2*4}(x-1{)}^2+\frac{1}{3*8}(x-1{)}^3-$cdots$+\frac{1}{n*2^n}(x-1{)}^n+$cdots

a$.$ As described in the statement of the Ratio Test, let $r_n(x)$ be the ratio of the $(n+1)$ st term of $T(x)$ to the $n$th term of $T(x).$ Find the simplest formula that you can for $r_n(x).$

b$.$ Let $r(x)=\underset{n}{lim}{r}_n(x).$ Evaluate this limit to find the simplest formula you can for $r(x).$

c$.$ For what values of $x$ is What does this tell us about $T(x)$ for these $x-$values?

d$.$ Let $T_{10}(x)$ be the sum of the first $10$ terms of $T(x),$ and let $f(x)=ln(2)-ln(3-x).$ Plot $f(x)$ and $T_{10}(x)$ on the same coordinate axes. What do you notice? What does this suggest about the series $T(x)?$