By David R. Morrison, Janos Kolla Summer Research Institute on Algebraic Geometry
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Additional resources for Algebraic Geometry Santa Cruz 1995: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz (Proceedings of Symposia in Pure Mathematics) (Pt. 2)
Solution All that happens is that the definition (1) remains valid with a possibly different value of X having to be chosen. This is often expressed in the following memorable way: ‘a finite number of terms do not matter’. For example, if 7 þ p will serve as a suitable value for X, then so will 12 or 37; but 10 might not. We have to prove that for each positive number ", there is a number X such that 1 < "; for all n > X: n3 (2) In1 order to find a suitable value of X for (2) to hold, we rewrite the inequality 3 < " in various equivalent ways until we spy a value for X that will suit n our purpose.
4. Problem 8 Prove that 4n > n4, for n ! 5. n n 2 n2 1 2 3 4 5 2 1 4 4 8 9 16 16 32 25 This assumption is just P(k). Since P(k þ 1) is: 2kþ1 ! (k þ 1)2. 1: Numbers 20 Three important inequalities in Analysis Our first inequality, called Bernoulli’s Inequality, will be of regular use in later chapters. Theorem 1 Bernoulli’s Inequality For any real number x ! À1 and any natural number n, (1 þ x)n ! 1 þ nx. Remark The value of this result will come from making suitable choices of x and n for particular purposes.
It follows that the original inequality ab is also true, for a, b 2 R. & 2 Remark pﬃﬃﬃﬃﬃ In the form ab aþb 2 this inequality is sometimes called the Arithmetic–Geometric Mean Inequality for a, b. A close examination of the above chain of equivalent statements shows that in À Á2 fact ab ¼ aþb if and only if a ¼ b. 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃ Problem 3 Prove that aþb a2 þ b2 ; for a, b 2 R. 2 pﬃﬃﬃ Problem 4 Suppose that a > 2. Prove the following inequalities: À Á À À ÁÁ2 (a) 12 a þ 2a < a; (b) 12 a þ 2a > 2: Hint: In part (b), use the result of Example 3 and the subsequent remark.