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26    MATHEMATICAL THEORY OF INVESTMENT
This result, which is obtained by taking four terms of the series,
is true to the fifth decimal place.
 Another expression which could not be easily computed by ordi-
nary means, and which occurs frequently in practice, is (1.04)".
BY the binomial series      1/1   \
                            —l——l|
      C1.04)A = 1 + ^ .04 +1^——(.04)-
                    JJl_lVl-2^
                    ^12__^12__'      _
                          1.2.3
Four terms of the series give the result
                   (1.04) ^= 1.003273 +,
which is accurate to the sixth decimal place.
  10. The exponential series.  The expression
                        limfl+^V                (1)
                        «=«\   n/
occurs so frequently and is of such importance in mathematics
that it may well be called a fundamental limit.
  To find an expression for it that will be useful in practical
work, suppose that n approaches infinity through the series of
natural numbers, i.e. through the positive integers taken in order
of increasing magnitude.  For any positive integral value of n

/-,  ,  x\"    -.  ,     a;  ,  n(n-V)x
1+- =l+w-+———o—-2
\   n/         n     1  2  n"

                      n(7t-l)(M-2)a-8     ^       ^
                         1.2.3   n3    n"'     ' 7
To find the value of the limit (1) by roughly approximate meth-
ods, divide out the n's from the numerators and denominators,
and C3) takes the form     -i
                       1--

(y\»   y   n „
1+^=1+^+-—^
   n/      1   1  -S
          /.   1\/.   2\

(i-lVi-
\  n/\

+^  M/v——^. a;^--- to n+1 terms. (3)
    1  2  3
                 LIMITS AND SERIES             27
If we assume that the limit of the sum on the right in (3) can be
found by taking the sum of the limits of the separate terms, even
though the number of terms be indefinitely increased, we shall have
            /    rX"      T    T2       Vs
        l-M^^r^r-hi-^-  <-)
The power series on the right is called the exponential series
and is ordinarily denoted by e*. By definition of e°,
             ^-^r^r-^---    <5)
                             
 The series (5) is convergent for all finite values of the vari-
able x, for the general term is
                         x"

and the test ratio is

1. 2.3...re

1- 2- 3--- n (w+1) _  x
        a:"                      n +1

                      1. 2. 2---n
                        X          1
Consequently,  t = lim ——- = x lim ——- = x 0.
           J     n=»ra+l    n=»w+l
The number t is therefore zero for every finite value of a;, and
the statement is proved.
 It is not difficult to prove that the expression e" behaves ex-
actly like an ordinary power when combined with other expres-
sions of the same sort.  In particular,
       ecev=ec'^\  e'— e" = e:c~v,  and  (^y = (PCV.
 To find the value of e1 or e, substitute 1 for x in the series (5).
Then              .,    ^      ..
              ^-^r-^r-ir^---
Ten terms of the series will give
                        e == 2.71827,
accurate to the fifth decimal place. The student is advised to
carry out the computation.