The present value of A due in 1 year is AR-1; the present value of A due in 2 years is AR-?; the present value of A due in 3 years is AR-3; and so on. [Art. 424.] Now V is the sum of the present values of the different payments; ... V=AR-1+AR-2+AR-3+......to n terms 1-R 1-R-1 R-1 Note. This result may also be obtained by dividing the value of M, given in Art. 426, by Rn. [Art. 423.] Cor. If we make n infinite we obtain for the present value of a perpetual annuity V= A A EXAMPLES XLI. 1. If in the year 1600 a sum of $1000 had been left to accumulate for 300 years, find its amount in the year 1900, reckoning compound interest at 4 per cent. per annum. Given log 104=2:0170333 and log 12885.5=4:10999. 2. Find in how many years a sum of money will amount to one hundred times its value at 52 per cent. per annum compound interest. Given log 1055=3.023. 3. Find the present value of $6000 due in 20 years, allowing compound interest at 8 per cent. per annum. Given log 2= •30103, log 3=.47712, and log 12875=4.10975. 4. Find at what rate per cent. per annum $1200 will amount to $20000 in 15 years at compound interest. Given log 2= •30103, log 3=.47712, and log 12063=4.08145. 5. Find the amount of an annuity of $100 in 15 years, allowing compound interest at 4 per cent. per annum. Given log 1.04=•01703, and log 180075=5.25545. 6. What is the present value of an annuity of $1000 due in 30 years, allowing compound interest at 5 per cent. per annum ? 7. A man borrows $5000 at 4 per cent. compound interest; if the principal and interest are to be repaid by 10 equal annual instalments, find the amount of each instalment. Given log 1.04=.01703, and log 675565=5.8296. CHAPTER XLII. LIMITING VALUES AND VANISHING FRACTIONS. 428. It will be convenient here to introduce a phraseology and notation which the student will frequently meet with in his mathematical reading. 429. An expression which involves any quantity, as x, and whose value is dependent on that of x, is called a function of x. Functions of x are usually denoted by symbols of the form f(x), F(x), +(x). Thus the equation y=f(x) may be considered as equivalent to a statement that any change made in the value of x will produce a consequent change in y, and vice versâ. The quantities x and y are called variables, and are further distinguished as the independent variable and the dependent variable. An independent variable is a quantity which may have any value we choose to assign to it, and the corresponding dependent variable has its value determined as soon as the value of the independent variable is known. 430. DEFINITION. If y=f(x), and if when x approaches a value a, the function f(x) can be made to differ by as little as we please from a fixed quantity b, then b is called the limit of y when x=a. For instance, if S denote the sum of n terms of the series 1 1 1 1 1 Here S is a function of n, and can be made as small as we please by increasing n; that is, the limit of S is 2 when n is infinite. 1+3+22+ 28 2n-1° 2n-1 ...... 431. We shall often have occasion to deal with expressions consisting of a series of terms arranged according to powers of some common letter, such as do taqx+Q2x2 + azx3 + ...... where the coefficients Qo, ay, A2, A3, are finite quantities independent of x, and the number of terms may be limited or unlimited. It will therefore be convenient to discuss some propositions connected with the limiting values of such expressions under certain conditions. 432. The limit of the series à+ax+a,x2 + a2x8+ when x is indefinitely diminished is ao. ppose that the series consists of an nfinite number of terms. Let b be the greatest of the coefficients aq, , Qg, ...; and let us denote the given series by ag+S; then S<bx+6x2+bx3 + ...; bx and if x<1, we have 1-2 Thus when x is indefinitely diminished, S can be made as small as we please; hence the limit of the given series is do.. If the series consists of a finite number of terms, S is less than in the case we have considered, hence still more is the proposition true. 433. In the series ao+a,x+a,x2 +azx8+ by taking x small enough we may make any term as large as we please compared with the sum of all that follow it; and by taking x large enough we may make any term as large as we please compared with the sum of all that precede it. The ratio of the term aniwn to the sum of all that follow it is an An+12+ An+222 + When x is indefinitely small the denominator can be made as small as we please; that is, the fraction can be made as large as we please. S< or Anxn + An+22n+2+ ... an Again, the ratio of the term anxn to the sum of all that precede it is anan " i An-12n-1 + An-2001–2+ An-ly+An-2y2 + 1 where or y= When x is indefinitely large, y is indefinitely small; hence, as in the previous case, the fraction can be made as large as we please. 434. The following particular form of the foregoing proposition is very useful. In the expression Anan +an-jan-1+ +ajx + 20 consisting of a finite number of terms in descending powers of X, by taking x small enough the last term a, can be made as large as we please compared with the sum of all the terms that precede it, and by taking x large enough the first term anan can be made as large as we please compared with the sum of all that follow it. Example 1. By taking n large enough we can make the first term of nt - 5n3 — 7 n +9 as large as we please compared with the sum of all the other terms; that is, we may take the first term n4 as the equivalent of the whole expression, with an error as small as we please provided n be taken large enough. 3x3 - 2x2-4 Example 2. Find the limit of when (1) is infi 5x3 — 4x+8 nite; (2) x is zero. (1) In the numerator and denominator we may disregard all 3X3 3 terms but the first; hence the limit is 5 -4 1 (2) When x is indefinitely small the limit is or 528' or VANISHING FRACTIONS. 435. Suppose it is required to find the limit of x2 + ax – -2a2 22 - a2 when x=a. = x + 2a If we put x=a+h, then h will approach the value zero as x approaches the value a. Substituting ath for x, x2 + ax — x - 2a? 3ah+h? 3ath X2 - a 2ah+h22ath' 3 and when h is indefinitely small the limit of this expression is 2 There is however another way of regarding the question; for x2 + ax – 2a2 (x-a)(x+2a) 3 and if we now put x=a the value of the expression is as before. If in the given expression x2 + ax -2a2 2 - X2 - a2 we put x=a before simplification it will be found that it assumes the form the value of which is indeterminate [Art. 168]; also we see that it has this form in consequence of the factor x—a appearing in both numerator and denominator. Now we cannot divide by a zero factor, but as long as x is not absolutely equal to a the factor x– a may be removed, and we then find that the nearer 2 approaches to the value a, the nearer does the value of the 3 fraction approximate sto or in accordance with the definition 2 of Art. 430, 3 when x=a, the limit of x2 + ax — 2a is X2 - a2 2 436. If f(x) and (x) are two functions of x, each of which becomes equal to zero for some particular value a of x, the fraction f(a) 0 takes the form (a) o' Example 1. If x=3, find the limit of 23 — 5x2+73-3 23 — 32 – 5x -3° When x=3, the expression reduces to the indeterminate form 0 ; but by removing the factor X-3 from numerator and denomi X2–2x+1 nator, the fraction becomes When x=3 this reduces 22 +29 +1 1 to which is therefore the required limit. , and is called a Vanishing Fraction. |