Page 172 - Calculus Demystified

Transcendental Functions
CHAPTER 6
[Refer to the “You Try It” following Example 5.9 in Subsection 5.2.3 for a 159
consideration of this limit.]
This formula tells us that, for large values of n, the expression
n
1
1 +
n
gives a good approximation to the value of e. Use your calculator or computer to
check that the following calculations are correct:
n
1
n = 10 1 + = 2.5937424601
n
n
1
n = 50 1 + = 2.69158802907
n
n
1
n = 100 1 + = 2.70481382942
n
n
1
n = 1000 1 + = 2.71692393224
n
n
1
n = 10000000 1 + = 2.71828169254.
n
With the use of a sufficiently large value of n, together with estimates for the error
term

n
1
e − 1 +
,
n
it can be determined that
e = 2.71828182846
to eleven place decimal accuracy. Like the number π, the number e is an irrational
number. Notice that, since exp(1) = e, we also know that ln e = 1.

EXAMPLE 6.13
Simplify the expression
5
ln(e · 8 −3 ).
SOLUTION
We calculate that
5
5
ln(e · 8 −3 ) = ln(e ) + ln(8 −3 )
= 5ln(e) − 3ln 8
= 5 − 3ln 8.