Activity 8.4.3

Let $f(x) = e^x$. 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 $e^x$.

1. For $f(x) = e^x$, explain why $f^{(k)}(0) = 1$ for every natural number $k$.

2. State the Taylor series, $T_f(x)$ centered at $a = 0$ for $f(x) = e^x$. Write $T_f(x)$ in both sigma notation and as an expanded sum.

3. Let $r_n(x)$ be the ratio of the $(n+1)^{\text{st}}$ term to the $n^{\text{th}}$ term of $T_f(x)$. Find the simplest expression you can for $r_n(x)$.

4. Let $r(x) = \lim_{n \to \infty} r_n(x)$. 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.

5. Use a computational device to graph $f(x) = e^x$, $T_{10}(x)$, and $T_{20}(x)$ on the same axes. What do you observe?