Activity 8.4.3.
Let \(f(x) = e^x\text{.}\) Our goal is to understand why the Taylor series for \(f(x)\) converges for every real number \(x\) and see that the Taylor series converges to \(e^x\text{.}\)
(a)
(b)
State the Taylor series, \(T_f(x)\) centered at \(a = 0\) for \(f(x) = e^x\text{.}\) Write \(T_f(x)\) in both sigma notation and as an expanded sum.
(c)
Let \(r_n(x)\) be the ratio of the \((n+1)^{\text{st}}\) term to the \(n^{\text{th}}\) term of \(T_f(x)\text{.}\) Find the simplest expression you can for \(r_n(x)\text{.}\)
(d)
Let \(r(x) = \lim_{n \to \infty} r_n(x)\text{.}\) Evaluate this limit, and then apply the Ratio Test to say what you can conclude about the \(x\)-values for which \(T_f(x)\) converges.
(e)
Use a computational device to graph \(f(x) = e^x\text{,}\) \(T_{10}(x)\text{,}\) and \(T_{20}(x)\) on the same axes. What do you observe?
