MATH_3333_HW5_solns - Math 3333 Homework 5 Solutions Name Peoplesoft ID Show your work If a problem requires a proof explain and justify your steps

MATH_3333_HW5_solns - Math 3333 Homework 5 Solutions Name...

This preview shows page 1 - 5 out of 12 pages.

Math 3333 Homework 5 Solutions Name: Peoplesoft ID: Show your work. If a problem requires a proof, explain and justify your steps carefully. Homework papers should be legible and neat, and the pages should be stapled together in the correct order. Illegible work may not be graded. Homework should be submitted in class on the indicated due date. Submissions by email or to the math office will not be accepted. 1. TRUE/FALSE: If the statement is true, prove it. If the statement is false, give a counter-example (a) If s n 0, then given any positive number there corresponds a positive integer N such that s n < for all n > N . True: s n 0 means that for any > 0, there is a positive integers N , such that, for all n > N , | s n - 0 | = | s n | < - < s n < . In particular, s n < for all n > N .
Image of page 1
(b) If for each positive number there is a positive integer N such that s n < for all n > N , then s n 0. False: Let s n = - n. Note that s n 6→ 0. ( s n diverges to -∞ .) However, for each > 0, we have s n = - n < 0 < for all n > N = 1. (c) If s n s and s n > 0 for all n , then s > 0. False: Let s n = 1 n . Then, s n > 0 for all n , but s n s = 0.
Image of page 2
(d) If ( s n ) and ( t n ) are divergent sequences, then ( s n + t n ) is divergent. False: Let s n = ( - 1) n and t n = ( - 1) n +1 . That is, ( s n ) = ( - 1 , 1 , - 1 , 1 , . . . ) ( t n ) = (1 , - 1 , 1 , - 1 , . . . ) Then, s n + t n = 0 for all n . Therefore, s n diverges, t n diverges, and s n + t n 0. (e) If ( s n ) and ( s n + t n ) are convergent sequences, then ( t n ) is convergent. True: Suppose s n s and s n + t n r . Then, by Theorem 1.2, Section 17, t n = [ s n + t n ] - s n r - s .
Image of page 3
(f) If ( s n ) is a sequence such that ( n s n ) converges, then s n 0.
Image of page 4
Image of page 5

You've reached the end of your free preview.

Want to read all 12 pages?

  • Fall '08
  • Staff
  • Math, Tn, Natural number, Sn

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture