Activity 8.4.2.
Consider the infinite Taylor series given by
\begin{align*}
T(x) \amp= \sum_{k = 1}^{\infty} \frac{1}{k \cdot 2^k}(x-1)^k\\
\amp= \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{align*}
(a)
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 \(T(x)\text{.}\) Find the simplest formula that you can for \(r_n(x)\text{.}\)
(b)
Let \(r(x) = \lim_{n \to \infty} r_n(x)\text{.}\) Evaluate this limit to find the simplest formula you can for \(r(x)\text{.}\)
(c)
For what values of \(x\) is \(|r(x)| \lt 1\text{?}\) What is the open interval of convergence for \(T(x)\text{?}\)
(d)
Let \(T_{10}(x)\) be the sum of the first \(10\) terms of \(T(x)\text{,}\) and let \(f(x) = \ln(2) - \ln(3-x)\text{.}\) Plot \(f(x)\) and \(T_{10}(x)\) on the same coordinate axes in a window centered around \(x=1\text{.}\) What do you notice? What does this suggest about the series \(T(x)\text{?}\)
