What is the significance of the limit laws for sequences?

[alingy1]

Hello everyone,
I'm starting to study sequences.
I'm on Stewart's Calculus textbook (single variable, 7th edition, for those who have it, on p. 693).
Now, I'm at the part where the limit laws are "transferred" to sequences.

(I'm sorry. I do not know how to code. I hope this is clear to you.)

lim as n->infinity of a_n^p=[lim as n->infinity of a_n]^p
IF p>0 AND a_n>0

Why do they add the IF p>0 AND a_n>0? I do not see why the formula would be wrong without that.

 

[PeroK]

Can you think of a counterexample if p = -1?

What could happen to a_n that would prevent the formula from being true?

Also, if a_n altenates between +1 and -1, can you see a counterexample?

 

[alingy1]

Why would a_n alternating change anything?
I tried to think about what would happen, but everything still makes sense.
Can you give me the counterexample you are thinking of? I'd like to graph it.

 

[PeroK]

If \ a_n = -1^n

Then what is:

lim_{n→∞}a_n

 

[alingy1]

lim of a_n as n->infinity is -1
lim of (a_n)^(-3) as n->infinity is -1
(lim of a_n as n->infinity)^(-3) is -1
The law seems to work! I think I'm doing something clearly wrong.

 

[PeroK]

If \ a_n = -1^n

Then what is:

lim_{n→∞}a_n

In this case, the limit does not exist. The sequence alternates between 1 and -1 doesn't get stay close to either. So:

(lim_{n→∞}a_n)^2 does not exist

But:

lim_{n→∞}a_{n}^2 = 1

 

[alingy1]

Did you mean a_n=(-1)^n? I think if a_n=-1^n, the limit is -1.

 

[PeroK]

Why would it be -1 in preference to 1? The sequence goes:

-1, 1, -1, 1, -1, 1, -1, 1...

Why would that converge to -1?

 

[alingy1]

(lim_{n→∞}a_n)^2 does not exist

But, if a_n=-1^n, the limit does exist!

 

[vela]

Did you mean a_n=(-1)^n? I think if a_n=-1^n, the limit is -1.

Yes, PeroK meant the former. The expression -1^n is a bit ambiguous as some people will read it as -(1^n) and others will read it as (-1)^n.

 

[alingy1]

Awesome! Thanks a lot. I'm starting to get this. Can I extrapolate that the a_n being >0 only adresses the issue of signs? What about the fact that p>0? I searched the web and no document explains it. Videos online, including KhanAcademy does not seem to address the issue!

 

[PeroK]

For the other one, try a_n = n and p = -1:

lim_{n→∞}a_{n}^{-1} = lim_{n→∞}\frac{1}{n} = 0

But:

lim_{n→∞}a_{n} = lim_{n→∞}n does not exist

so:

(lim_{n→∞}n)^{-1} is not defined

 

[vela]

lim as n->infinity of a_n^p=[lim as n->infinity of a_n]^p
IF p>0 AND a_n>0

Why do they add the IF p>0 AND a_n>0? I do not see why the formula would be wrong without that.

PeroK already gave you a counterexample with p=2 and $a_n = (-1)^n$. In that case, $a_n^p = 1$, so the limit on the lefthand side exists. The limit on the righthand side, however, doesn't.

What about if p=1/2 but $a_n$ wasn't necessarily positive?

What about if p=-1 and $a_n \to \infty$?

What about p=0 and $a_n=n$?

Can you see why the equality wouldn't hold in each of these cases?

 

[alingy1]

Thanks PeroK!
Vela:
What about p=0 and a_n=n? This seems to work:
Left-hand limit=1
Right-hand limit=1

 

[vela]

Really? What's $\lim_{n \to \infty} n$?

 

[alingy1]

Infinity, no?

 

[vela]

Better to say the limit doesn't exist.

 

[alingy1]

True, but doesn't the answer give infinity^0=1?

 

[vela]

No. You can't raise something that doesn't exist to any power.

 

[PeroK]

True, but doesn't the answer give infinity^0=1?

I'm sure the book you have will help with this, but one of the best things you can do before you go much further is:

1) Understand why ∞ is not a number. And, in particular, that no arithemtic operations are defined on it.

2) Understand precisely what we mean by n → ∞ and why this does not require ∞ to be a number.

This is very important before you go very far with real analysis.