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\noindent{\scriptsize
  \href{http://www.math.toronto.edu/~drorbn/Copyleft}{\textcopyleft} $\mid$
  \href{http://www.math.toronto.edu/~drorbn}{Dror Bar-Natan}:
  \href{http://www.math.toronto.edu/~drorbn/classes}{Classes}:
  \href{http://www.math.toronto.edu/~drorbn/classes/#1213}{2012-13}:
  \href{http://drorbn.net/index.php?title=12-267}{267 Advanced ODEs}:
}
\newline
\vskip 1mm
\parpic[r]{\parbox{1.1in}{\centering
  \includegraphics[width=1.1in]{Lazarus_Immanuel_Fuchs.jpg}
  \tiny Lazarus Immanuel Fuchs, 1833--1902
}}
{\LARGE Fuchs' Theorem}

\vskip 1mm

{\scriptsize Following Taylor's {\em Introduction to Differential Equations}.}

\begin{theorem} Suppose the series $v(x)=\sum_{k=0}^\infty v_kx^k$ solves the $n$-dimensional system $v'(x)=A(x)v(x)+g(x)$, where $A(x)$ and $g(x)$ are given by power series $A(x)=\sum_{k=0}^\infty A_kx^k$ and $g(x)=\sum_{k=0}^\infty g_kx^k$ that converge at radius $R$ for some $R>0$. Then the series $v(x)$ converges for any $x$ with $|x|<R$.
\end{theorem}

\begin{proof} \picskip{2}
Below $||M||$ where $M$ is a matrix or a vector means ``the largest absolute value of an entry of $M$''.

The convergence of the series for $A$ and for $g$ implies that there are constants $\alpha$ and $\gamma$ such that
\[ ||A_k||<\alpha R^{-k} \qquad\text{and}\qquad ||g_k||<\gamma R^{-k}. \]
We wish to show that whenever $r<R$, there is a constant $\eta$ such that
\begin{equation} \label{vbound} ||v_j||<\eta r^{-j}. \end{equation}
This we shall do by the method of ``induction with an undetermined hypothesis''. Namely, we assume that for some $k$ Equation~\eqref{vbound} holds for all $j\leq k$, without specifying $\eta$. We then prove that~\eqref{vbound} is true for $j=k+1$ and see what conditions this may put on $\eta$. We keep track of these conditions, and at the end of the proof we verify that we could have satisfied them at the start of the proof.

The equation $v'=g+Av$ implies that $(k+1)v_{k+1}=g_k+\sum_{j=0}^k A_{k-j}v_j$. Therefore
\begin{multline*}
  (k+1)||v_{k+1}||\leq ||g_k||+\sum_{j=0}^k ||A_{k-j}v_j||
  \leq ||g_k|| + n\sum_{j=0}^k ||A_{k-j}||\cdot||v_j|| \\
  < \gamma R^{-k} + n\sum_{j=0}^k \alpha R^{j-k}\cdot\eta r^{-j}
  = \gamma R^{-k} + n\alpha\eta r^{-k}\sum_{j=0}^k \left(\frac{r}{R}\right)^{k-j}.
\end{multline*}
The last sum is a geometric sum with ratio smaller than $1$. Hence its value is bounded by some fixed constant $\beta$. Hence
\[ (k+1)||v_{k+1}|| < \gamma R^{-k} + \alpha\eta n\beta r^{-k} < r^{-k}(\gamma+\alpha\eta n\beta), \]
and thus, assuming $\eta\geq\gamma$,
\[ ||v_{k+1}|| < r^{-(k+1)}\frac{r(\gamma+\alpha\eta n\beta)}{k+1}
  \leq \eta r^{-(k+1)}\frac{r(1+\alpha n\beta)}{k+1}. \]
Now for large enough $k$, say for $k>K$, the ugly fraction in the last formula will be smaller than $1$, and we will have proven Equation~\eqref{vbound} for $j=k+1$. We still need to make sure that Equation~\ref{vbound} holds for $j\leq K$. But this places only finitely many conditions on $\eta$, so we just need to pick $\eta$ so that
\[ \eta > \max\left(\gamma, r^j||v_j||\right)_{j\leq K}. \]
\end{proof}

\noindent{\footnotesize
  Dror Bar-Natan, \today;
  \url{http://drorbn.net/index.php?title=12-267}.\newline
  Sources at \url{http://drorbn.net/AcademicPensieve/Classes/12-267/}.
}

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