help needed 216 and 227 Defnition 21b for the for of d Produ

help needed: 2.1.6 and 2.2.7.

Defnition 2.1.b for the for of (d) Produce a definition similar to that l.b the case that sequence diverges to -oo. (e) Use your definition to show that the sequences (-r) and (n-n3) d n3) diverge to 0o Exercise 2.1.6. (a) Suppose that (a) and (b) are sequences with (a)-a. Shou (b) Show that it is possible for two sequences (an) and (b.) to both diverge 2.1.6, (a) Suppose that (an) and (bn) are sequences with (an) that if (an-b.) 0, then (b.) a. diverge even if (an-b.) 0.

Solution

2.1.6

a.

For e/2>0 there exist N1 so that

|a_n-a|<e/2 for all n>N1

For e/2>0 there exists N2 so that

|an-bn|<e/2 for all n>N2

Let N =max{N1,N2}

So for all n>N

|bn-a|=|(bn-an)+(an-a)|<|bn-an|+|an-a|<e/2+e/2=e

Hence, bn converges to a

b)

Let, an=n,bn=n

both diverge to infinity

By an-bn =0 which converge to 0

2.2.7

a)

an converges to a so for e>0 there exist N so that for all n>N

|an-a|<e

||an|-|a||<=|an-a|<e for all n>N

Hence |an| converges to |a|

(b)

Consider the sequence,

a_{2n}=1,n=1,2,...

a_{2n+1}=-1,n=0,1,2,...

So even terms are equal to 1 and odd terms are -1

So magnitude of the terms is equal to 1

Hence, |an| converges to 1

But an does nto converge to 1 or -1

c)

|an| converges to 0

So for e>0 there exist N so that for all n>N

||an||<e for all n>N

But ||an||=|an|

|an|<e for all n>N

Hence an converges to 0