if x and y are proper fractions and positive, and x > y. 1 1 But х ..), log +2 (1+++), [Art. 226]; 1-x 3 5 ... *261. To prove that (1 + x)1+* (1 − x)1-*>1, if x<1, and to log P = (1+x) log (1 + x) + (1 − x) log (1 − x) = x {log (1 + x) — log (1 − x)} + log (1 + x) + log (1 − x) Hence log P is positive, and therefore P>1; 3. Shew that the sum of the mth powers of the first n even numbers is greater than n (n+1)m, if m> 1. 4. If a and ẞ are positive quantities, and a> α B, shew that 5. n lies between 2 and If a, b, c are in descending order of magnitude, shew that 9. If a, b, c are in H. P. and n > 1, shew that an+cn>2bn. 1 1 10. Find the maximum value of x3 (4a-x)5 if x is positive and less than 4a; and the maximum value of x2(1-x)3 when r is a proper fraction. 11. If x is positive, shew that log (1+x) < and > 1+x 12. If x+y+2=1, shew that the least value of and that (1-2) (1-y) (1-z)>8xyz. and 1 1 1 13. Shew that (a+b+c+d) (a3+b3+c3+d3)>(a2+b2+c2+d2)2. Shew that the expressions 14. a (a - b) (a-c)+b (b−c) (b− a) +e(c-a) (e-b) are both positive. 15. Shew that (xm+ym)n < (x2+yn)m, if m>n. 17. If a, b, c denote the sides of a triangle, shew that (2) a2yz+b2zx+c2xy cannot be positive, if x+y+z=0. 18. Shew that 1 3 5 19. If a, b, c, d, ...... are p positive integers, whose sum is equal to n, shew that the least value of where q is the quotient and the remainder when n is divided by p. CHAPTER XX. LIMITING VALUES AND VANISHING FRACTIONS. α Ꮳ 262. Ir a be a constant finite quantity, the fraction can be made as small as we please by sufficiently increasing x; that is, we can make approximate to zero as nearly as we please a Ꮳ by taking a large enough; this is usually abbreviated by saying, "when x is infinite the limit of is zero." α α x Again, the fraction - increases as x decreases, and by making α x as small as we please we can make as large as we please; х thus when x is zero has no finite limit; this is usually ex α pressed by saying, "when x is zero the limit of 263. When we say that a quantity increases without limit or is infinite, we mean that we can suppose the quantity to become greater than any quantity we can name. Similarly when we say that a quantity decreases without limit, we mean that we can suppose the quantity to become smaller than any quantity we can name. The symbol is used to denote the value of any quantity which is indefinitely increased, and the symbol 0 is used to denote the value of any quantity which is indefinitely diminished. 264. The two statements of Art. 262 may now be written symbolically as follows: But in making use of such concise modes of expression, it must be remembered that they are only convenient abbreviations of fuller verbal statements. 265. The student will have had no difficulty in understanding the use of the word limit, wherever we have already employed it; but as a clear conception of the ideas conveyed by the words limit and limiting value is necessary in the higher branches of Mathematics we proceed to explain more precisely their use and meaning. 266. 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+ + + + ; then S=2. ... 2 22 23 Here S is a function of n, and as we please by increasing n; n is infinite. that is, the limit of S is 2 when 267. 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 where the coefficients a,, α, α, αz, 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. |