Activity 8.4.2

Consider the infinite Taylor series given by $$\begin{aligned} T(x) &= \sum_{k = 1}^{\infty} \frac{1}{k \cdot 2^k}(x-1)^k \\ &= \frac{1}{1 \cdot 2}(x-1) + \frac{1}{2 \cdot 4}(x-1)^2 + \frac{1}{3 \cdot 8}(x-1)^3 + \cdots + \frac{1}{n \cdot 2^n}(x-1)^n + \cdots \end{aligned}$$

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

2. Let $r(x) = \lim_{n \to \infty} r_n(x)$. Evaluate this limit to find the simplest formula you can for $r(x)$.

3. For what values of $x$ is $|r(x)| \lt 1$? What is the open interval of convergence for $T(x)$?

4. 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 in a window centered around $x=1$. What do you notice? What does this suggest about the series $T(x)$?