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\begin{document}
\pset{5}{March 3, 1998}{Tuesday, March 10}

{\bf Reading assignment}: Rosen 1.7--8; 6.042 Fall97 Notes for
Lectures 8 and 9.

\begin{problems}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 1
\problem 

A college loan officer offers prospective M.Eng students the following
deal.  He will give them \$23,100 at the start of each year for five years
starting today to pay for their college tuition.  He offers the following
plan for repayment:\\
Wait five years, then repay \$$x.00$ at the start
of each year for the next ten years.

Suppose the annual interest rate is 8\%.  What value of $x$ will make the 
present value of the repayment equal to the present value of the loan?

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 2 (Rosen 1.7, 11 and 12)

\problem \bparts

\ppart Find a closed form solution for \[\sum_{k=1}^n {1
/k(k+1)}\] \hint Use the fact that ${1 / k(k+1)}=1 / k - 1 / (k+1)$.

\ppart Use induction to verify your closed form.
\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 3
\problem 
Find a closed form solution for \[\prod_{i=1}^n 2 \cdot 4^i\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 4
\problem 

 	Let $0< x < 1$ and $0 < y < 1$.  Evaluate the following infinite sums.
\bparts
\ppart $$\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} x^i y^{i+2j}.$$
\ppart $$\sum_{i=1}^{\infty} \sum_{j=1}^{\infty} ij x^i y^j.$$
\eparts


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 5
\problem 
Justify your answers to the following:
\bparts
\ppart Is $2^n = O(4^n)$? 
\ppart Is $2^{2n} = O(2^n)$? 
\ppart Is $2^{n+1} = O(2^n)$?
\ppart Is $\log_{10} x = \Theta (\log_2 x)$?
\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 6
\problem 
Find a $\Theta$ bound for $\sum_{k=1}^n k^6$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 7
\problem 

Prove or give a counterexample: If $f(n) = \Theta(n)$, then $\sum_{i=1}^n f(i)
= \Theta(n^2)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 8
\problem 
\bparts
\ppart Find a closed form solution for
$\sum_{i=1}^n i^2x^i$.
\ppart Verify your closed form by induction.
\eparts

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Problem 9
\problem 

Let $S_n = \sum_{i=1}^n i^{1 /3}$.  Use the integral method to determine
\[\lim_{n \rightarrow \infty} \frac{S_n}{n^{4 / 3}} \]


\end{problems}

\end{document}

