1873_solutions

# Is a number when we say that b a f x dx is convergent

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Unformatted text preview: j! 2 jn1 Using the fact that k lim x1 jn1 2j ! j x j j! 2 4 k 4j jn1 2j ! j! 2 we choose a number t  1 such that k jn1 4j 2j ! j t  1. j! 2 We therefore know that Ý sup j1 n 2j ! j x 4 j j! 2 j1 2j ! j x 4 j j! 2 Ý 0 x1 jn1 2j ! j t  1. 4 j j! 2 x log x n converges uniformly in x on the interval 0, 1 . 6. Prove that the series We begin by observing that x log x 0 as x 0 (from the right). Now the expression x log x takes its minimum value when log x  1  0, in other words, when x  1 . Therefore the maximum value of e |x log x | is 1 and the fact that e 1 x log x n en for every positive integer n and every number x 0, 1 allows us to use the comparison test to x log x n converges uniformly in x on the interval 0, 1 . deduce that 7. Given that f n and g n are sequences of real valued functions defined on a set S, that f and g are functions defined on S and that f n f and g n g pointwise as n Ý, prove that a. f n  g n f  g pointwise as n Ý. Given any x S, the fact that f n x fn x  gn x f x g x . b. f n gn c. f n g n f g pointwise as n fg pointwise as n d. In the event that g x f x and g n x g x as n Ý guarantees that Ý. Ý. 0 for every number x in the set S we have f n /g n f/g pointwise as n Ý. 8. Given that f n and g n are sequences of real valued functions defined on a set S, that f and g are functions defined on S and that f n f and g n g boundedly as n Ý, prove that a. f n  g n b. f n c. f n g n gn f  g boundedly as n Ý. f Ý. g boundedly as n fg boundedly as n Ý. d. In the event that there exists a number   0 such that |g n x | 362  for each n and every number x in the set S we have f n /g n f/g boundedly as n Ý. These assertions follow at once from Exercise 7 after we have observed that if f n and g n are bounded sequences of functions then so are f n  g n etc. 9. Suppose that f n is a sequence of real valued functions defined on a set S and that f is a given function defined on S. Prove that the following conditions are equivalent: a. The sequence f n converges uniformly to the function f on the set S. b. For every number holds for all n  0 there exists an integer N such that the inequality sup|f n f |  N.  0 there exists an integer N such that the inequality |f n x f x |  holds for all n N and all x S. Conditions a and b are obviously the same and it is clear that these imply condition c. To show that condition c implies condition b, suppose that  0. Using the fact that 2 is a positive number, choose an integer N such that the inequality |f n x f x |  2 holds whenever n N and x S. Then for all n N we have . sup|f n f | 2 c. For every number 10. Suppose that f n is a sequence of real valued functions defined on a set S and that f is a given function defined on S. Examine the following two conditions:  0 there exists an integer N such that the inequality |f n x f x |  N and all x S. For every number holds for all n For every number holds for all n  0 and every number x S there exists an integer N such that th...
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