c255exw01 - Final Examination Monday April 23 Mathematics...

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Final Examination Monday, April 23 Mathematics 189-255B 1. (i) (4 marks) Define the term metric space . (ii) (4 marks) Define the term open subset of a metric space. (iii) (4 marks) Define the term closed subset of a metric space. (iv) (8 marks) Show from first principles that a subset of a metric space is closed if and only if its complement is open. 2. For each of the following series, determine whether the series converges. Justify your answer. (i) (5 marks) X n =1 n + 1 - n . (ii) (5 marks) X n =1 sin π n 2 + 1 n . (iii) (5 marks) X n =1 3 n n ! n n . (iv) (5 marks) X n =3 (ln n ) - ln n . 3. (i) (4 marks) Define the term Riemann partition . (ii) (4 marks) Define the upper and lower Riemann sums U ( P, f ) and L ( P, f ) for a Riemann partition P . Let f : [0 , 1] -→ [ - 1 , 1] be defined by f ( x ) = ( - 1) k if x ]2 - ( k +1) , 2 - k ] and f (0) = 0. (iii) (8 marks) Given > 0 find explicitly a Riemann partition P of [0 , 1] with U ( P, f ) - L ( P, f ) < . Justify your answer. What is the significance of what you have just shown?
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