Math 53 – Math of Finance
Monday, January 22, 2007
Homework 1 --- Due Wednesday, January 24, 2007
Some Taylor series
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r = 0.01 |
r = 0.10 |
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1+r |
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1+r+r2/2 |
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1+r+r2/2+r3/3! |
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exp ( r ) |
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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 
.
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R = 0.01 |
R= 0.10 |
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R-R2/2 |
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R-R2/2+R3/3 |
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ln ( 1 + R ) |
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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)