28 MATHEMATICAL THEORY OF INVESTMENT 11. The logarithmic series. The series x2^ x4 ^ a;-^+3--^+ C1) is called the logarithmic series. It is convergent for values of .; which are numerically less than unity, for the general term is
and the test ratio is ——- — — = ——-- x. n+1 n n+i The limiting value of the test ratio is then (= lim ——- x B=»TC+1 = x lim ——— = x. '"l+l n The series (1) is therefore convergent for |(|=[a- <1, which was to be proved. The series (1) is called the logarithmic series because for values of x numerically less than 1, i.e. for values for which the series is convergent, it represents the value of the expression log (1 + x). By definition * log(l+^)=^-^+^-^+.... (2) The logarithmic series is not well adapted to computation, since it converges very slowly and a large number of terms are re- quired to obtain a good approximation. The actual work of computing logarithms is best accomplished by the use of some such series as that of Example 9 below, which is derived from the logarithmic series. * The logarithm that is here defined is the hyperbolic, or Napierian, logarithm. For a definition of different systems of logarithms, see the next chapter. LIMITS AND SERIES 29 EXAMPLES 1. Find the value of (l.Oo)8 to five decimal places. 2. Find the fifth root of 1.05, i.e. the value of (1 + .05)°, to five decimal places. 3.. Find the value of e~1 to five decimal places by means of the power series 2 -s (fc=l+l+Y-^T^-^- 4. If sin x is defined by the series jS 3.6 sin x = x - ——- + . - ,- . - - ,
1.2.3 1.2-3-4.5 find the value of sin (.1) to the fourth decimal place. 5. For what values of x is the series 1 \ x \ xlt \ xs \ \ x" i 1.2 2-3 3.4 4.5 (n +!)(»+ 2) convergent ? 6. For what values of x is the series 1+1+1+1+...+!+... convergent? \- x x x x 7. Prove by actual multiplication that the product of i. vl y3 ex=l+-+ ——+ —-— + 1 1.2 1.2.3 11ifi W^ ^1+^+^+... is e^y. 8. Compute the value of e3 to the fifth decimal place. 9. Compute the value of the logarithm of 2 from the series , M ^VM-N ,1 !M- JV\8 , 1 IM-N\6 , I logN=2[M^N+3{W+N) +1;(-M^N) + } by making M = 2 and ^V = 1.