Indeterminate Forms. -Vanishing Fractions. 389. When one or more variables are involved in both numerator and denominator of a fraction, it may happen that for certain values of the variables both numerator and denominator of the fraction vanish. The fraction then assumes If there is but one variable the indeterminate form 0 0 involved, we may obtain a definite value as follows: Let x be the variable, and a the value of x for which the 0 fraction assumes the form Give to x a value a little 0 greater than a, as a + h; the fraction will now have a definite value. The limit of this last value, as h is indefinitely decreased, is called the limiting value of the fraction. The fundamental indeterminate form is o ee indeterminate forms may be reduced to this. and all other 0 as x approaches a. When x has the value a, the fraction assumes the form Since h is not 0, we can divide by h and obtain 2 a + h. As h is indefinitely decreased, this approaches 2 a as a limit. 2. Find the limiting value of 2x-4x+5 when x 1 becomes infinite. Divide each term of the numerator and denominator by x3. Then, As x increases indefinitely, each term that contains x of the last fraction approaches 0 as a limit (Theorem 4), and the fraction approaches as a limit. CHAPTER XXIV. PROPERTIES OF SERIES. Convergent and Divergent Series. 390. By performing the indicated division, we obtain the infinite series 1 + x + x2 + x3 + ..... 1 from This 1 x series, however, is not equal to the fraction for all values of x. х 391. If x is numerically less than 1, the series is equal to the fraction. In this case we can obtain an approximate value for the sum of the series by taking the sum of a number of terms; the greater the number of terms taken, the nearer will this approximate sum approach the value of the fraction. The approximate sum will never be exactly equal to the fraction, however great the number of terms taken; but by taking enough terms it can be made to differ from the fraction by as little as we please. 1 Thus, if x = the value of the fraction is 2, and the series is , 1 1 1 1+ + + + 2 4 8 The sum of four terms of this series is 17; the sum of five terms, 118; the sum of six terms, 13; and so on. The successive approximate sums approach, but never reach, the finite value 2. 392. An infinite series is said to be convergent when the sum of the terms, as the number of terms is indefinitely increased, approaches some fixed finite value; this finite value is called the sum of the series. ..... 393. In the series 1 + x + x2 + x3 + suppose x numerically greater than 1. In this case the greater the number of terms taken, the greater will their sum be; by taking enough terms we can make their sum as large as we please. The fraction, on the other hand, has a definite value. Hence, when x is numerically greater than 1, the series is not equal to the fraction. Thus, if x = 2, the value of the fraction is series is 1+2+4+8+ 1, and the The greater the number of terms taken, the larger the sum. Evidently, the fraction and the series are not equal. 394. In the same series suppose x = 1. In this case the The more terms we take, the greater will the sum of the series be, and the sum of the series does not approach a fixed finite value. If x, however, is not exactly 1, but is a little less than 1, Suppose x = 1. In this case the fraction is 1+1 2' If we take an even number of terms, their sum is 0; if an odd number, their sum is 1. Hence, the fraction is not equal to the series. 395. A series is said to be divergent when the sum of the terms, as the number of terms is indefinitely increased, either increases without end, or oscillates in value without approaching any fixed finite value. No reasoning can be based on a divergent series; hence, in using an infinite series it is necessary to make such restrictions as will cause the series to be convergent. Thus, we can use the infinite series 1 + x + x2 + x3 + when, and only when, x lies between + 1 and 1. ..... 396. THEOREM. If two series, arranged by powers of x, are equal for all values of x that make both series convergent, the corresponding coefficients are equal each to each. For, if A+B x + Сx2 + .. A' + B'x + C'x2 + ·····, by transposition, A — A' = (B' — B) x + (С" − С') x2 + Now, by taking a sufficiently small, the right side of this equation can be made less than any assigned value whatever, and therefore less than A- A', if A- A' has any value whatever. Hence, A A' cannot have any value. or = Therefore, A A' 0, or A Hence, Bx + Сx2 + Dx3 + = = A'. B'x+C'x2+ D'1⁄23 + · (B- B') x Divide by x, B — B' = (C' — C) x + (D' − D) x2 + ·· By the same proof as for A In like manner, A', B – B' = 0, or B = B'. C C', D= D'; and so on. Hence, the equation if true for all finite values of x, is an identical equation; that is, the coefficients of like powers of x are equal. |