24 MATHEMATICAL THEORY OF INVESTMENT Five terms of this series give cos (1)=.540302, which is correct to the fifth decimal place. Further examples are given at the end of the chapter. Among the most important power series are the binomial series, the exponential series, and the logarithmic series. These three will be considered in the order named. 9. The binomial series. In elementary algebra we are made familiar witli the expansion of a power of a binomial when the exponent is a positive integer. Such an expansion has the form (a +xr= a" +^a"l^+M(^.~21) a""v+ +xn- ^ This expansion is a power series consisting of a finite number of terms with coefficients ,,„ n i M(re-l) , ^ a, .a , ———— a , , j., and whose general term is n(n-l)(n-2)---(ra-r.+l) ^_y ,9. 1.2. 3-..(r-l)r It may also be looked upon as a power series all of whose coeffi- cients after the (r+l)st are zero. If the exponent n is not a positive integer, it is still possible to obtain a development for (a + x)" which will be similar in form to (1) and for which the general term will be (2). There is this great difference, how- ever : when n is not a positive integer, the series will never end but is a true infinite series. The power series a" + na^x + n (M -1) ffl-'a-2 + (3) -L ^ is called the binomial series. To determine the character of the binomial series, consider the test ratio, viz. H(H-l)(M-2)---(n-r+l)^-y 1 2-3---(r—l)r _______n—r+1 x n(7i-l)(n-2)...(ra-r+2) , ,~ r 'a 1.2.3... (r-1) LIMITS AND SERIES The limit of the test ratio is
25
t= lim
n— r+1a;
x,. n—r+1 = - hm —————
But in order that the series (3) should be convergent, ( must be less in absolute value than unity, i.e. y — - < 1, or | r < | a \. C4~) a 111 \ ^ If, on the other hand, |a-| > |a|, the series (3) will be divergent. We write (a + x)" =an+ na"-^ + M<M-1) a-V +-.. 1 A for all values of n, with the understanding that the power series represents the development of (a+x)" for those values of x which satisfy the convergence condition (4). A case of special importance is that for which a = 1; then (1+.)'=1+^+M^)^+M<?^-2)+... (5) for values of x such that \x\ < 1. The series (5) is of great use in computing many quantities which occur in the theory of interest. As a simple example, sup- pose the value of the expression —— is required. This value may readily be obtained by replacing a: by .04 in the binomial expansion,