smidterm1sol

smidterm1sol - CALCULUS 153: SAMPLE MIDTERM 1 SOLUTIONS...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the 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 < } . 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 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 wont 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

Page1 / 4

smidterm1sol - CALCULUS 153: SAMPLE MIDTERM 1 SOLUTIONS...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online