Concepts Assignment
 
1. Read Chapters 1 and 2 of Golub and Ortega. 

2. An important attitude in the computational sciences is that the 
        computer is to be used as a tool of exploration and discovery.  
        The computer should be used to check out ``hunches'' or 
        conjectures, which then later should be subjected to analysis in 
        the traditional manner.  There are limits to this approach, 
        however.  An example is in limiting processes.  Because the 
        computer deals with finite quantities, the results of a 
        computation may be misleading.  Explore each of the situations 
        below, using C or Fortran.  A few minutes or even seconds of 
        computing should be enough to give you a feel for the nature  
        of the computations.  Be careful not to tie up the 
        computer too long. 

          a.  Consider the question of the convergence of the series, 

                 \sum_{i=1} i. 

                Obviously, this series does not converge.  Suppose, 
                however, that we begin summing this series using 
                floating-point numbers.  Will the series overflow?  If 
                so, at what value of $i$ (approximately)?  State your 
                answer in terms of the standard parameters of the 
                floating-point model, $b$, $p$, $e_{\rm min}$, and  
                $e_{\rm max}$.  Or will the series converge?  If so, to 
                what value, and at what value of $i$ (approximately)? 
          b.  What about the harmonic series, 

                  \sum_{i=1} 1/i? 

          c. What about 

                  \sum_{i=1} 2^{-2i}? 

   Write out your answers to this question in a file in your directory.  
   Put a link to it in your home page.