is of importance. Hence we prove as follows: I. Its value is less than 3. II. Since it is less than 3, the series is convergent. III. Its value is 2.71828182... This may be easily calculated [See Ex. I. (1)]. IV. It is incommensurable. For suppose that it is commensurable; it can then be put where m and n are integers. In this case Multiply each side of this supposed identity by [n then mn 1 = a whole number + 1 1 + + etc. n + 1 (n+1)(n+2) is a proper fraction; for it is greater than n+1 + + etc. 1 and less than = a whole number + a proper fraction; which is absurd. V. Since the numerical value of the series is incommensurable, and we know of no surd or other algebraical expression that is equal to it, it is usual to express its numerical value by the letter e. [cf. E. 28.] 1 + etc., i. e. less than n 1 (a whole number) EXAMPLES. I. (1) Calculate the value of e by taking the first 13 terms of the series. (2) Prove that the first 13 terms of the series will give the value of e correct to 9 significant figures. + + + +etc. =e. 13 5 17 (6) Prove that the series x-x2+}x3 − x2+etc. is con - + 2 vergent if x is greater than 1 and is not greater than 1. 2. Expansion of e* in ascending powers of x. Since nx (1 + 1) = {(1 + 1)*}* always; and since by the binomial theorem This statement is arithmetically intelligible and true provided both these series are convergent. They are convergent for all values of n greater than 1. Therefore they are arithmetically intelligible and true however great n may be. And in the limit, when n is infinitely increased, the above statement becomes [cf. Art. 8] or, 1 +1+ + {1+ 1 |3 e2 = 1 + + etc. х + } This result is called the Exponential Theorem. 3. To expand a" in ascending powers of x. Then Let a be any number, and let c = loga, so that e =α. THE LOGARITHMIC EXPANSION. 4. In the above expansion put 1+y for a, and we This may be put into a different form thus: (1 + y)* − 1 Ꮳ = х + etc. · loge (1 + y) + 12 {loge (1 + 3)}" + terms con taining higher powers of x =loge (1+ y) + x, R, where R is a quantity which is not infinite when x = 0. The limit of the right-hand side when x = 0 is loge (1 + y). and this, when x = 0, has for its limit y — {y3 + }y3 — {y1 + etc. This series is convergent when y is equal to or numerically less than 1. Therefore, when y lies between -1 and +1 or is equal to 1, 2 loge (1+ y) = y − 1 . y2 + } . y3 − 1 . y* + etc. This is the required Logarithmic Expansion. EXAMPLES. II. (1) Calculate the numerical value of twelve terms of the and calculate the value of log, 2 to 2 decimal places. 1 1 (4) 1 (Result '69...) + + + etc.=2 logey - loge (y+1) - loge (y − 1). y2 2y 3ys 1 (6) 2 {27+1+3(3x+1)+5 (24+1) + etc.} |