Spring2002 Midterm

# Spring2002 Midterm - MATH 138 Spring 2002 Midterm Page 1 of...

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MATH 138 — Spring 2002 — Midterm Page 1 of 2 Reprinted by MathSoc ./ Instructors: M. Shpigel, S. Sivaloganathan, R. Andre 1. (a) (3 marks) State the comparison theorem for improper integrals. (b) (5 marks) Show that Z e 1 x (ln x ) 2 dx converges and ﬁnd its value. (c) (4 marks) Does the integral Z 1 1 x 3 + x dx converge? Justify your answer. 2. Evaluate where possible, with justiﬁcation , the following limits. (a) (3 marks) lim t 27 t 1 / 3 - 3 t - 27 (b) (3 marks) lim x 0 + x ln x (c) (3 marks) lim x →- 1 x 2 - 1 x + 1 (d) (3 marks) lim x 0 cos 2 x - cos 3 x x 2 3. (a) (4 marks) Deﬁne what is meant by the following: i. The sequence { a n } is bounded below . ii. The sequence { a n } is decreasing . iii. The sequence { a n } converges . (b) (6 marks) Let { a n } be a sequence deﬁned by a 1 = 3 and a n +1 = 1 4 - a n . Show that { a n } is decreasing and 0 < a n 3. (c) (4 marks) Explain why { a n } has or hasn’t a limit L . If it does, then compute L . 4. (a) (4 marks) Deﬁne what is meant by the statement: “the series X n =1 a n is convergent.” (b) (6 marks) Determine which of the following series are convergent or divergent. Justify your answers. i. X n =1 1 e 2 n . ii. X n =1 ln

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## This note was uploaded on 10/21/2010 for the course MATH 138 taught by Professor Anoymous during the Fall '07 term at Waterloo.

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Spring2002 Midterm - MATH 138 Spring 2002 Midterm Page 1 of...

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