Math 53 – Math of Finance

Monday, January 22, 2007

Homework 1 --- Due Wednesday, January 24, 2007

 

Some Taylor series

 

r = 0.01

r = 0.10

1+r

 

 

1+r+r2/2

 

 

1+r+r2/2+r3/3!

 

 

exp ( r )

 

 

1.  The series for    is   .   The series converges absolutely for every  .

 

            a.  Compute    and its Taylor approximations

                        for    and   ŕ

 

            b.  Show that    whenever  .

 

                                                                                                [ Note: exp(r) is a synonym for er. ]

 

2.  In this class, log and ln both mean natural logarithm.  One series for  ln  is

            .  It converges absolutely whenever .

 

R = 0.01

R= 0.10

R-R2/2

 

 

R-R2/2+R3/3

 

 

ln ( 1 + R )

 

 

 

            a.  Compute    and its Taylor

                        approximations for    

                        and    ŕ

 

            b.  Show that    whenever

                          and  .

 

An Exponent

 

3.  Let  a  be any positive constant.  Then   for all  .

            Express its derivative as    

            where the blank represents a constant depending on  .

 

Some limits

4.         (Take  or ; try L’Hospital or binomial theorem.)

5.            ( x is any real constant. )

6.          ( x and y are any real constants. )

7.      ( T and x are positive constants; = lower-case Greek  tau.)

 

(end)