CHAPTER XXXV. EXPONENTIAL AND LOGARITHMIC SERIES. 357. The Exponential Series is an infinite series whose first term is 1 and whose (n+1)th term is "/n !. It is absolutely convergent for all finite values of x [Art. 323, Ex. 2]. Let its limit be denoted by f(x); then the two exponential series may be multiplied together and their product will be a convergent series [Art. 324]. Compare the terms of this product with the terms of the series The coefficient of ayn in f(x) xf(y) is clearly 1/(m!n!) and in f(x+y) the term (x+y)+n/(m + n)! is the only one in which "y" can occur, and by the binomial theorem for a positive integral index, the coefficient of xy" in (x + y) "+"/(m + n)! is Hence, whatever x and y may be, the coefficients of xmy" in f(x) ×f(y) and in f(x + y) are the same, however large m and n may be, that is, whatever (integral) values, between 1 and +∞, m and n may have. It follows that the series (iii.), with its terms expanded, is identical with the series formed as the product of the series (i.) and (ii.); hence f(x)׃(y)=f(x+y), for all values of x and y. By repeated application of (iv.) we obtain ... f(x) ׃(y) × f(t) × ··· = f(x+y) × ƒ(t) × ... = f(x+y+t+........). Therefore, if z be any positive integer, ƒ(1) × ƒ(1) × ƒ(1) × to z factors =ƒ(1+1+1+... to z terms), (iv.) If z be a positive fraction, say z=p/q, where p and q Hence e2 = f(z), for all positive values of z both integral and fractional. If z be negative, say zz', then by (iv.), f(−x)xf(*)=f(0), and f(0) is obviously equal to 1. Hence, for all rational values of z, be they integral or fractional, positive or negative, Should be irrational, we replace it by a rational fraction, which may be so chosen as to differ from the given irrational value by an arbitrarily small quantity. [See Art. 249.] 358. We may now derive an infinite series, in ascending powers of x, for the exponential function a*. of which a can be expanded in a series of ascending is sometimes called the exponential theorem. powers of x, = 359. The Logarithmic Series. Let a e, so that In a a2 = e2m2 = 1 + x ln a + 1, (x lna)2 + 1, (xlna) 3 + ... Now put a = 2! 1+y; then we have {x 3! (1 + y)2 = 1 + xln(1 + y) + 1; {æ ln(1+y) }2 + ........ 2! Now, provided y be numerically less than unity, (1+y)* can be expanded by the binomial theorem; we then have 1 =1+xln (1+ y) + 2; {x ln(1 + y) }2 + ........ 2! ; Equate the coefficients of x on the two sides of the last equation. [Art. 330.] We thus obtain y2 y3 y1 In(1+y)=y- + + 2 3 4 This is called the logarithmic series. 360. In order to diminish the labour of finding the approximate value of the logarithm of any number, more rapidly converging series are obtained from the funda mental logarithmic series. Changing the sign of y in the logarithmic series y2 y3 3 |