stochastic calculus

# stochastic calculus - JOMO KENYATTA UNIVERSITY OF...

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JOMO KENYATTA UNIVERSITY OF AGRICULTURE AND TECHNOLOGY YEAR III SEMESTER I EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE IN FINANCIAL ENGINEERING, ACTURIAL SCIENCE AND STATISTICS STA 2306: REAL ANALYSIS FOR STATISTICS DATE: AUGUST 2013 TIME: 2 HOURS Instructions: Answer Question ONE and any other TWO questions QUESTION ONE (30 MARKS) a) State any two conditions for the convergence of Fourier series (2 marks) b) Evaluate the following limit lim ( x , y ) ( 0 , 0 ) h xy 3 x 2 + 3 y 2 i (4 marks) c ) State whether the following functions are even or odd i. f ( x ) = x 2 cos x ii. f ( n ) = n 2 tan n (4 marks) d) If f ( x , y ) = cos3 x sin4 y find f x and f y at ( π 12 , π 6 ) (5 marks) e) Let n = 1 a n and n = 1 b n be convergent series, show that n = 1 a n + b n converges (5 marks) f) Given that the series n = 1 1 n diverges, use the limit comparison test to determine the convergence of n = 1 sin 1 n (5 marks) g) Let f ( x ) be defined by the infinite series n = 1 nx n , show that f ( x ) = x ( 1 - x ) 2 (5 marks) 1

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