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\newcommand{\myCourseNumber}{Math 142B}
\newcommand{\myAssignmentDueDate}{11:00pm Thursday, June 6, 2024}
\newcommand{\myAssignmentTitle}{Homework Assignment 5}
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\begin{questions}

%%Ross 23.5
\question Let $\sum a_n x^n$ be a power series with radius of convergence $R$. Prove:
\begin{parts}
\part If all the coefficients $a_n$ are integers and $a_n \not= 0$ for infinitely many $n$, then $R \leq 1$.
\part If \ $\limsup |a_n| > 0$, \ then $R \leq 1$.
\end{parts}

%%Ross 23.6
\question \begin{parts}
\part Suppose $\sum a_n x^n$ has finite radius of convergence $R$ and $a_n \geq 0$ for all $n$. Show that if the series converges at $x = R$, then it also converges at $x = -R$.
\part Exhibit an example of a power series whose interval of convergence is exactly $(-1, 1]$. \newline  (\textbf{Note:} ``Exhibit'' means ``Show that the example has the required properties.'')
\end{parts}

%%Ross 26.4
\question \begin{parts}
\part Verify that \ $\displaystyle e^{-x^2} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^{2n}$ \ for all $x \in \mathbb{R}$, since \ $\displaystyle e^x  = \sum_{n=0}^\infty \frac{1}{n!} x^n$ \ for all $x \in \mathbb{R}$.
\part Write \ $\displaystyle F(x) = \int_0^x e^{-t^2} \, dt$ \ as a power series.  Be sure to briefly explain how you know that the power series for $F(x)$ converges for all $x \in \mathbb{R}$.
\end{parts}

%%Ross 26.5
\question Let \ $\displaystyle f(x) = \sum_{n=0}^\infty \frac{1}{n!} x^n$ \ for $x \in \mathbb{R}$.  Using only the properties of power series, show that $f' = f$.

%%Ross 26.6
\question For $x \in \mathbb{R}$, let
\begin{align*}
s(x) &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n+1}}{(2n+1)!} \\
c(x) &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots = \sum_{n=0}^\infty (-1)^n \frac{x^{2n}}{(2n)!}
\end{align*}
Prove:
\begin{parts}
\part $s' = c$ \ and \ $c' = -s$.
\part $\displaystyle \left( s^2 + c^2 \right)' = 0$.
\part $\displaystyle s^2 + c^2 = 1$.
\end{parts}

%%Ross 26.8
\question \begin{parts}
\part Show that \ $\displaystyle \sum_{n=0}^\infty (-1)^n x^{2n} = \frac{1}{1 + x^2}$ \ for $x \in (-1,1)$.
\part Show that \ $\displaystyle \arctan(x) = \sum_{n=0}^\infty \frac{(-1)^n}{2n + 1} x^{2n + 1}$ \ for $x \in (-1,1)$.
\part Show that the equality in (b) also holds for $x = 1$.  Use this to find a fun formula for $\pi$.
\part What happens at $x = -1$?
\end{parts}

%% Ross 31.1
\question Find the Taylor series for $\cos(x)$ and indicate why it converges to $\cos(x)$ for all $x \in \mathbb{R}$.

%% Ross 31.5
\question Let $g(x) = \begin{cases} 0 & \text{if} \enspace x = 0, \\ e^{-1/x^2}  & \text{otherwise}. \end{cases}$
\begin{parts}
\part Show that \ $g^{(n)}(0)$ \ for all $n \in \mathbb{N}$.
\part Show that the Taylor series for $g$ about $0$ agrees with $g$ only at $x = 0$.
\end{parts}

%% Fitzpatrick 8.2.5
\question Prove that \enspace $\displaystyle \left| \sin(x + h) - \left(\sin(x) + h \cos(x)\right) \right| \leq \frac{h^2}{2}$ \enspace for every pair of real numbers $x$ and $h$.

%% Ross 31.11
\question Suppose $f$ is differentiable on $(a,b)$, \ $f'$ is bounded on $(a,b)$, \ $f'$ never vanishes on $(a,b)$, \ and the sequence $(x_n)$ in $(a,b)$ converges to $\bar{x} \in (a,b)$.

Show that if \ $\displaystyle x_n = x_{n-1} - \frac{f(x_{n-1})}{f'(x_{n-1})}$ \ for all $n \geq 0$, \ then \ $f(\bar{x}) = 0$.

\end{questions}

\end{document}
