This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CALCULUS 153: SAMPLE MIDTERM 1 SOLUTIONS Problem 1 (16 points) . Determine the least upper bound and greatest lower bound of the following sets, or state that they do not exist. You do not need to justify your answer (4 points each). (1) (∞ , 3] ∪ [4 , 7) , (2) { x : 2 < ln( x + 1) < 4 } , (3) { a n } where a n = 2 1 /n , (4) { x ∈ Q : e < x < π } . Solution (1) The lub is 7, the glb does not exist. (2) The lub is e 4 1, the glb is e 2 1. (3) The lub is 2, the glb is 1 (note that the sequence is decreasing and converges to 1). (4) The lub is π and the glb is e . Note that for any real number a there are rational numbers arbitrarily close to a ; therefore, specifying that x must be a rational won’t affect the lub and glb. Problem 2 (20 points) . For each of the following sequences, determine whether the sequence converges or diverges. If it converges, find its limit. Show your work. (5 points each). (1) a n = 2 n + ( 1) n n ! ; (2) b n = 1 + x n 2 n ; (3) c n = sin( n 2 + ln n ) n 1 /n n + 1 ....
View
Full
Document
This test prep was uploaded on 04/07/2008 for the course MATH 153 taught by Professor Masson during the Fall '07 term at UChicago.
 Fall '07
 Masson
 Calculus, Sets

Click to edit the document details