give an example of each of the following, or state that such a request is imposs

Question

give an example of each of the following, or state that such a request is impossible by referencing the proper theorem(s):

(a) sequences ( xn )and (yn ), which both diverge, but whose difference xn - yn converges;

(b) bounded sequences ( xn ) and (yn ), where (xn ) converges, ( yn ) diverges, and ( xn + yn ) diverges;

(c) a divergent sequence bn with   bn not equal to 0 for all n such that (1/bn ) diverges

(d) an unbounded sequence ( an ) and a divergent sequence ( bn ), with ( a n - b n ) convergent;

(e) two sequences (a n ) and ( bn ), where ( an ) and ( bn ) diverge but ( anbn ) does not.
  

Explanation / Answer

a)

xn=yn=n

xn-yn=0

b)

xn=1,yn=(-1)^n

xn+yn=1+(-1)^n, diverges

c)

bn=sin(n), diverges

1/bn =1/sin(n) also diverges

d)

an=bn=n

an-bn=0

e)

an=bn=(-1)^n

anbn=1

Related Questions

Navigate

• Browse (All)
• Browse G
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.