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Math 188, Spring 2021

Homework 2

Due: April 16, 2021 11:59PM via Gradescope

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Solutions must be {\bf clearly} presented. Incoherent or unclear solutions will lose points.

\begin{enumerate}
\item 
  \begin{enumerate}
  \item Let $(A_i(x))_{i \ge 0}$ be a sequence of formal power series such that $\lim_{i \to\infty} A_i(x) = A(x)$. Let $\sigma$ be a permutation of the non-negative integers.

    Prove or find a counterexample: $\lim_{i \to \infty} A_{\sigma(i)}(x)$  converges to $A(x)$. 
  \item Suppose we are given a doubly-indexed sequence of formal power series $(A_{i,j}(x))_{i,j \ge 0}$ such that:
    \begin{itemize}
    \item For each $i$, the limit $\lim_{j \to \infty} A_{i,j}(x)$ converges to $B_i(x)$ and $\lim_{i \to \infty} B_i(x)$ converges to $B(x)$, and
    \item For each $j$, the limit $\lim_{i \to \infty} A_{i,j}(x)$ converges to $C_j(x)$ and $\lim_{j \to \infty} C_j(x)$ converges to $C(x)$.
    \end{itemize}
    Prove or find a counterexample: $B(x) = C(x)$
  \end{enumerate}
  
\item Let $F(x)$ be a formal power series with $F(0)=0$.
  \begin{enumerate}    
  \item Show that there exists a formal power series $G(x)$ with $G(0)=0$ such that $F(G(x))=x$ if and only if $[x^1]F(x) \ne 0$.
  \item Assuming $[x^1]F(x) \ne 0$, show that $G(x)$ is unique and also satisfies $G(F(x))=x$.

    You may use without proof that composition of formal power series is associative.
  \end{enumerate}

\item Assume now that we deal with complex-coefficient formal power series.
  Define the following sets of formal power series:
  \begin{align*}
    V &= \{F(x) \mid F(0)=0\},\\
    W &= \{G(x) \mid G(0)=1\}.
  \end{align*}
  \begin{enumerate}
  \item Given $F \in V$, show that $\bR(F) = \sum_{n \ge 0} \frac{F(x)^n}{n!}$ is the {\it unique} formal power series $G \in W$ such that $DG(x) = DF(x) G(x)$. This defines a function $\bR \colon V \to W$.

    [Conventions: $F(x)^0=1$ even if $F(x)=0$ and $0!=1$]
  \item Given $G \in W$, show that there is a {\it unique} formal power series $F \in V$ such that $DF(x) = DG(x)/G(x)$. We define the function $\bL \colon W \to V$ by $\bL(G) = F$.

    [For the rest, it is unnecessary to use explicit formulas for $\bL$ and $\bR$ and in fact it may be easier to only use the uniqueness properties above.]
  \item Show that $\bR$ and $\bL$ are inverses of each other.
  \item Show that $\bR(F_1+F_2) = \bR(F_1) \bR(F_2)$ for all $F_1,F_2 \in V$.
  \item Show that $\bL(G_1G_2) = \bL(G_1) + \bL(G_2)$ for all $G_1,G_2 \in W$.  
  \item If $m$ is a positive integer and $G \in W$, show that $\bR(\frac{\bL G}{m})$ is an $m$th root of $G$. (Hence this gives an alternative proof for the existence of $m$th roots.)
    

  \end{enumerate}
\item Let $n \ge 2$ be an integer.
  \begin{enumerate}
  \item Prove that
    \[
      \sum_{i=0}^n i \binom{n}{i} (-1)^{i-1} = 0.
    \]
  \item Compute
    \[
      \sum_{\substack{0 \le i \le n\\i \text{ even}}} i \binom{n}{i}.
    \]
  \end{enumerate}

  \section{Optional problems (don't turn in)}
\item Let $A_0(x), A_1(x), \dots$ and $B_0(x), B_1(x), \dots$ be sequences of formal power series. Assume that $\lim_{i\to\infty} A_i(x) = A(x)$ and $\lim_{i \to \infty} B_i(x) = B(x)$. Prove that
  \begin{align*}
    \lim_{i \to \infty} (A_i(x) + B_i(x)) &= A(x)+B(x),\\
    \lim_{i \to \infty} (A_i(x) B_i(x)) &= A(x)B(x).
  \end{align*}

\item Which of the following infinite products of formal power series converge? 
  \begin{enumerate}
  \item $\prod_{i \ge 0} (1+x^{i+1})$
  \item $\prod_{i \ge 0} (1+x)^{i+1}$
  \end{enumerate}
  
\item Give proofs for the following identities for formal derivatives of formal power series (for the second identity, assume $A(0)=0$ so that the compositions are defined):
  \begin{align*}
    D(A \cdot B) &= (DA) \cdot B + A \cdot (DB),\\
    D(B \circ A) &= (DA) \cdot (DB \circ A).
  \end{align*}

\item Let $A(x)$ be a formal power series. Define its {\bf order} ${\rm ord}(A)$ to be the smallest $k$ such that $[x^k]A(x) \ne 0$. Show that there exists a formal power series $B(x)$ such that $B(x)^2=A(x)$ if and only if the order $k$ of $A(x)$ is even, and our field of scalars contains a square root of $[x^k]A(x)$.
  
\item Use the notation from (3).
  \begin{enumerate}
  \item Show that if $\sum_{i \ge 0} F_i(x)$ converges to $F(x)$, then $\prod_{i \ge 0} \bR(F_i)$ converges to $\bR(F)$.
  \item Show that if $\prod_{i \ge 0} G_i(x)$ converges to $G(x)$, then $\sum_{i \ge 0} \bR(G_i)$ converges to $\bR(G)$.
  \end{enumerate}
  
  \item I am not sure how easy the following is to prove without knowing the right trick, but I'm curious if you can come up with a nice solution.

    Suppose we pick a sequence of numbers $(a_2,a_4,a_6,\dots)$. Show that there is a unique way to complete it to a sequence $(a_n)_{n \ge 2}$ such that $A(x) = -x + \sum_{n \ge 2} a_nx^n$ satisfies $A(A(x))=x$.

  \end{enumerate}

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