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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Discussion 3 -- Using the ``Big Limit Theorem''}
\centerline{September 20, 2000}
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\noindent
{\it Background}
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Last time we proved a ``Big Limit Theorem'' showing how convergent
sequences can be combined to produce other convergent sequences using
the algebraic sum and product operations on ${\bf R}$.
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\proclaim Theorem.  Let $x_n \to L$ and $y_n \to M$ as $n \to \infty$.
Then 
\item{a)}  $x_n \pm y_n \to L\pm M$,
\item{b)}  $x_n\cdot y_n \to L\cdot M$, and
\item{c)}  if $M \ne 0$, and $y_n \ne 0$ for all $n \in {\bf N}$, then
${x_n\over y_n} \to {L\over M}$.
\par

\noindent
It is also true, for instance, that if $x_n \ge 0$ for all $n$ and
$x_n\to L$, then $\sqrt{x_n} \to \sqrt{L}$ as $n \to \infty$.


One of the very nice features of these results is how they give 
{\it justifications} for the ``intuitive'' approaches we have used to 
determine limits (e.g. by rearranging the formula for the $n$th term
in various algebraic ways).  In this discussion, we will see how 
that works, then look at some additional facts that can be derived
from the "Big Limit Theorem".  
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\noindent
{\it Discussion Questions}
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\item{A)}  For each of the following convergent sequences, use one or more 
parts of the ``Big Limit Theorem'' to find the limit, and explain how
you are using the theorem.  You may use without proof ``standard facts''
such as $\lim_{n\to\infty} {1\over n} = 0$, or $\lim_{n\to\infty} c^n = 0$
if $|c| < 1$.
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\itemitem{1)}  $x_n = {4n^3 + 2\over n^3 + n}$
\itemitem{2)}  $y_n = {2^n \over 1.5^n + 2.5^n}$ (Hint:  Try to 
identify the fastest growing term, then rearrange algebraically to 
``neutralize'' it. )
\itemitem{3)}  $z_n = \sqrt{4n^2 + n} - \sqrt{4n^2 - 3}$ 
(Hint:  The algebraic fact $(\sqrt{a} - \sqrt{b})(\sqrt{a} + \sqrt{b}) = a - b$
may be useful here.)
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\noindent
\item{B)}  In this problem you will develop a useful sufficient condition
for when $x_n \to 0$ based on examining the {\it ratios}
between successive terms: ${x_{n+1}\over x_n}$.
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\itemitem{1)}  Assume that $x_n$ is a sequence of positive numbers
and that for a fixed $c < 1$, ${x_{n+1}\over x_n} < c$ for all $n \ge 1$.
Show (using one or more parts of the ``Big Limit Theorem'') that
$\lim_{n\to\infty} x_n = 0$.
\itemitem{2)}  Now assume that $x_n$ is a sequence
of positive numbers such that $\lim_{n\to\infty} {x_{n+1}\over x_n} = a < 1$.
Show that again $\lim_{n\to\infty} x_n = 0$.  (Hint:  
If $z_n$ is a sequence with $\lim_{n\to\infty} z_n = a < 1$ and
$a < c < 1$, then there exists an $n_0 \in {\bf N}$ such that $z_n < c$ 
for all $n \ge n_0$.  The idea is the same as in problem 6 from \S 2.1.)  
\itemitem{3)}  Use part 2 of this question to show that 
$$\lim_{n\to\infty} {b^n\over n!} = 0.$$
for all $b \in {\bf R}$.
\itemitem{4)}  Use part 2 of this question to show that 
$$\lim_{n\to\infty} {(n!)^2\over (2n)!} = 0.$$
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\noindent
{\it Assignment}
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Group writeups, due in class Monday, September 25.
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