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.)

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