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# HW 10 - MATH 290 HOMEWORK 10 SOLUTIONS In this homework we...

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MATH 290 – HOMEWORK 10 – SOLUTIONS In this homework, we will prove from scratch (that is, without using L’Hopital’s Rule) that lim n 1 /n = 1. 1. Prove that lim s 2 n - 1 = 0. Do this using the definition of convergence, and by invoking the Archimedean Principle of the real numbers explicitly where it is required. Solution: We need to show that given > 0 there is an N N such that if n N then s 2 n - 1 - 0 < . Let > 0. Using the Archimedean Principle of the real numbers, we can choose N N such that N > (2 / 2 ) + 1. If n N , then n N > (2 / 2 ) + 1, so that n > (2 / 2 ) + 1. From this it follows that n - 1 > 2 / 2 , that 1 / ( n - 1) < 2 / 2, that 2 / ( n - 1) < 2 , and finally that s 2 n - 1 = s 2 n - 1 - 0 < as required. 2. Prove the following by induction. (a) For every x > 0 and every n N , (1 + x ) n > nx . (b) For every x > 0 and every n N with n 2, (1 + x ) n > n ( n - 1) 2 x 2 . (Hint: You may use the result in part (a) for this proof, and the fact that ( n + 1) n - n ( n - 1) = 2 n .) Solution: (a). If n = 1 then the inequality reduces to 1 + x > x which is true. Hence the result holds for n = 1.

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HW 10 - MATH 290 HOMEWORK 10 SOLUTIONS In this homework we...

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