How Does the Behavior of k^n Change with Different Ranges of k?

[gregy6196]

Homework Statement

prove that if k>1 then kⁿ→∞ an n→∞

there is a hint given. (hint:let k=1+t where t>0 and use the fact that (1+t)ⁿ>1+nt)

(ii) prove that if 0<k<1 then kⁿ→0 as n→∞

Homework Equations

The Attempt at a Solution

if k>1
then kⁿ⁺¹-kⁿ=k^n(k-1)

(ii) 1/k>1
then1/kⁿ→∞

 

[SammyS]

Homework Statement

prove that if k>1 then kⁿ→∞ an n→∞

there is a hint given. (hint:let k=1+t where t>0 and use the fact that (1+t)ⁿ>1+nt)

(ii) prove that if 0<k<1 then kⁿ→0 as n→∞

Homework Equations

The Attempt at a Solution

if k>1
then kⁿ⁺¹-kⁿ=k^n(k-1)

This appears to have a typo in it. It should read:

kⁿ⁺¹-kⁿ=kⁿ(k-1)​

(ii) 1/k>1
then1/kⁿ→∞

For (i), try using the hint.

Should you be using an ε - δ argument for this ?

 

[gregy6196]

the hint doesn't make sense to me and i think it is ussing epsilon-delta

 

[SammyS]

...
Should you be using an ε - δ argument for this ?

Excuse me,

I believe that I should have said ε - N.

DUH !

 

[gregy6196]

yh ur right, but how would i approach this

 

[gregy6196]

Help!

 

[SammyS]

Pretty much what you need to show is that given any number, M > 0, (no matter how large) there exists a natural number, N, such kⁿ > M, for all n > N .

I presume you have some similar criterion for showing that lim_n→∞a_{n[/SUB= ∞ .}

 

[Ray Vickson]

the hint doesn't make sense to me and i think it is ussing epsilon-delta

What is it about the hint that is confusing you? Using the hing is by far the easiest way of doing (i). And, if you don't want to use an unsubstantiated hint, proving it is easy enough, just by induction (assuming t > 0, of course).

RGV