The Maclaurin series for $e^{x}$ is $e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \cdots + \frac{x^{n}}{n!} + \cdots$. The continuous function $f$ is defined by $f(x) = \frac{e^{(x-1)^{2}} - 1}{(x-1)^{2}}$ for $x \neq 1$ and $f(1) = 1$. The function $f$ has derivatives of all orders at $x = 1$. (a) Write the first four nonzero terms and the general term of the Taylor series for $e^{(x-1)^{2}}$ about $x = 1$. (b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 1$. (c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b). (d) Use the Taylor series for $f$ about $x = 1$ to determine whether the graph of $f$ has any points of inflection.
The Maclaurin series for $e^{x}$ is $e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + \cdots + \frac{x^{n}}{n!} + \cdots$. The continuous function $f$ is defined by $f(x) = \frac{e^{(x-1)^{2}} - 1}{(x-1)^{2}}$ for $x \neq 1$ and $f(1) = 1$. The function $f$ has derivatives of all orders at $x = 1$.
(a) Write the first four nonzero terms and the general term of the Taylor series for $e^{(x-1)^{2}}$ about $x = 1$.
(b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor series for $f$ about $x = 1$.
(c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b).
(d) Use the Taylor series for $f$ about $x = 1$ to determine whether the graph of $f$ has any points of inflection.