16. Shew that, if m be a positive integer, then 17. Shew that, if m, n and m n+1 are positive integers; + 1.2.3 to (n + 1) terms (m. 12 (m − n + 1 ) (m − n + 2) (m − n + 3) (m + 1) (m + 2)...(m + n) · − n + 1 ) (m − n + 2)... (m − n + m)' 1 m (m-1) (m-2). 5 ... 1 (m + 1) (m + 2) 19. Shew that, if P. be the sum of the products r together of the first n even numbers, and Q, be the sum of the products r together of the first n odd numbers; then will and 1 + P, + P2+...... + P = 1.3.5... (2n+1), 20. Prove that {a + (a + 1) + (a + 2) + ... + (a + n)} {a2 + (a + 1) + (a + 2) + ... ... + (a + n)} = a3 + (a + 1)3 + ... + (a + n)3. 21. Shew that the series n 1-a” (1−an) (1−a"− 1) __ (1−a”) (1 − a′′ − 1) (1 − a′′ − 2) is zero when n is an odd integer, and is equal to (1 − a) (1 − a3) ...(1-a") when n is an even integer [Gauss]. 25. Sum to infinity the series 1.2.3 +3.4.5x+5.6.7x2 +7.8.9x3 + x being less than unity. 26. Shew that, if n is a positive integer 27. Shew that, if a, a, a,...be all positive, and if a1 + a + a + ... be divergent, then 2 + a, +1 (a, + 1) (a ̧ + 1) + (a ̧ + 1) (a ̧ + 1) (α, + 1) + · 2 (a ̧ 3 2 + + 2m+x + ...+ 3m+x 4m+x (n + 1)m+x is convergent if x > 1, and is divergent if x 1. 29. Shew that, if the series u1+u2+u ̧ + divergent, the series + 30. For what values of x has the infinite product (1+a) (1+ax) (1 + ax3) (1 + ax3)... a finite value? 31. Prove that, if v is always finite and greater than unity but approaches unity without limit as n increases indefinitely, the two infinite products v ̧♥ ̧♥ ̧♥......., are either both finite or both infinite. 8 2 8 32. Test the convergency of the following series :— CHAPTER XXVI. INEQUALITIES. 339. WE have already proved [Art. 228] the theorem that the arithmetic mean of any two positive quantities is greater than their geometric mean. We now proceed to consider other theorems of this nature, which are called Inequalities. Note. Throughout the present chapter every letter is supposed to denote a real positive quantity. 340. The following elementary principles of inequalities can be easily demonstrated: I. If a b; then a +x>b+x, and a- x>b-x. II. If a > b; then a <- b. III. If a > b; then ma > mb, and — ma <- mb. IV. If a > b, a' > b', a" > b", &c; then a +a+a" +...>b+b' + b′′ +..., and aa'a"...> bb'b".... -m V. If a >b; then am > bm, and a ̄m <b ̄m. Ex. 1. Prove that a3+b3>a2b+ab2. We have to prove that a3 - a2b-ab2+b3>0, or that (a2 – b2) (a - b)>0, which must be true since both factors are positive or both negative according as a is greater or less than b. Ex. 2. Prove that am+a-m>an+a¬n, if m>n. We have to prove that (am – an) (1 − a−m-n)>0, which must be the case since both factors are positive or both negative according as a is greater or less than 1. Ex. 3. Prove that (12 + m2 + n2) (l'2 + m22 + n'2) > (ll' + mm' + nn')2. (12 + m2 + n2) (l'2 + m22 + n'2) − (ll' + mm' +nn')2 Now the last expression can never be negative, and can only be zero when mn' — m'n, nl' — n'l and lm' – l'm are all separately zero, the Hence (12 + m2 + n2) (l'2 + m22 + n'2) > (ll' + mm' + nn')2, except when l/l'=m/m'=n/n', in which case the inequality becomes an equality. 341. Theorem I. The product of two positive quantities, whose sum is given, is greatest when the two factors are equal to one another. For let 2a be the given sum, and let a + x and a be the two factors. Then the product of the two quantities is a2x2, which is clearly greatest when x is zero, in which case each factor is half the given sum. The above theorem is really the same as that of Art. 228; for from Art. 228 we have (a + b) (a + b) > > ab. 342. Theorem II. The product of any number of positive quantities, whose sum is given, is greatest when the quantities are all equal. For, suppose that any two of the factors, a and b, are unequal. Then, keeping all the other factors unchanged, take (a+b) and (a + b) instead of a and b: we thus, without altering the sum of all the factors, increase their continued product since (a + b) × 1 (a + b) > ab, except when a = b. Hence, so long as any two of the factors are unequal, the continued product can be increased without altering the sum; and therefore all the factors must be equal to one another when their continued product has its greatest possible value. |