Pure Mathematics

Fractal attractors by Grasberger

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There is no 1000-digit number that is equal to the sum of the 1000th powers of its digits). 3, we gave a cunning geometrical √ construction that demonstrated the existence of the real number n for any positive integer n. However, proving the existence of a cube root and, more generally, an nth root of any positive real number x is much harder and requires a deeper analysis of the reals than we have undertaken thus far. We shall carry out such an analysis later, in Chapter 24. 2, and state it here.

Xn ∈ R, and suppose that k of these numbers are negative and the rest are positive. If k is even, then the product x1 x2 . . xn > 0. And if k is odd, x1 x2 . . xn < 0. PROOF Since the order of the xi s does not matter, we may as well assume that x1 , . . , xk are negative and xk+1 , . . , xn are positive. 1, −x1 , . . , −xk , xk+1 , . . , xn are all positive. By (4), the product of all of these is positive, so (−1)k x1 x2 , . . , xn > 0 . If k is even this says that x1 x2 , . . , xn > 0.

1 Let x be a real number. n) (i) If x = 1, then x + x2 + x3 + ∙ ∙ ∙ + xn = x(1−x 1−x . (ii) If −1 < x < 1, then the sum to infinity x + x2 + x3 + ∙ ∙ ∙ = x . 1−x 21 A CONCISE INTRODUCTION TO PURE MATHEMATICS 22 PROOF (i) Let sn = x + x2 + x3 + ∙ ∙ ∙ + xn . Then xsn = x2 + x3 + ∙ ∙ ∙ + xn + xn+1 . Subtracting, we get (1 − x)sn = x − xn+1 , which gives (i). (ii) Since −1 < x < 1, we can make xn as small as we like, provided we take n large enough. So we can make the sum in (i) as close as we like to x 1−x provided we sum enough terms.

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