\item   % next time change to p_0 = 1/3 for easier analysis
Consider the following recursive equation
$p_0 = 1, \quad p_1 = a_1, \quad p_n = {10\over 3} p_{n-1} - p_{n-2}.$
For what values of $a_1$ is it stable in relative error? Explain.

%% \item
%% Give the rate of convergence for 
%% $ \displaystyle{ \sum_{n =1}^\infty }{1 \over n^2} = {\pi^2 \over 6} $
%% either analytically or numerically.

\item % Make sure Lim a_n = 0 next time (so that scaled plotting helps)
The file an.txt contains a sequence stored as '$n, a_n$' at $n$th line. 
Find its rate of convergence. Express your answer as $O(\beta_n)$ and
find $\beta_n$ explicitly. Show details. 

\item
Find a root of $x = 2 \cos x$ with 10 correct decimal digits
using any numerical method of your choice.
Put (1): the detail formula, (2): $x_0$, $x_1$ and (3): the answer $x^*$,
on the answer sheet, but need not hand in the code.

\item
Show that the nonlinear equation $x = 1 + \cos(x)/2$
has a solution in $[1, 1.5]$. 
Let $x_0=1.25$, give an estimate on $N$ (need not be optimal) such
that  $| x_n- x^* | < 10^{-5} $ for all $n \geq N$.

% \item % next time give up to p4 and find p5, correct determination of p5
      % requires knowledge on all previous steps.
% The first few iteration
% $(p_i, f(p_i))$, $i=0,1,2,3$ of method of false position for
% some equation $f(x) = 0$ is given in q2p5.txt.
% Find $p_4$ (4 digits will do). 
% Also give your formula for finding $p_4$ and explain.