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# Convergence or Divergence? Watch

1. Hi guys, I'm having some trouble showing that these sequences either converge or diverge. I can see intuitively that some have a limit or go off to infinity but I'm having trouble proving it mathematically...

I need to consider what happens as n tends to infinity...

1)

For this one, I can see that 4^n can be written as 2^(2n). As the numerator has a higher power than the denominator then the sequence will not have a limit. At least I think that is right

2)

In this one I'm a little confused due to the alternating term.
2. (Original post by DJB MASTER)
Hi guys, I'm having some trouble showing that these sequences either converge or diverge. I can see intuitively that some have a limit or go off to infinity but I'm having trouble proving it mathematically...

I need to consider what happens as n tends to infinity...

1)

For this one, I can see that 4^n can be written as 2^(2n). As the numerator has a higher power than the denominator then the sequence will not have a limit. At least I think that is right
No, that's not right. You can *always* change the power. E.g. .

2)
If tends to a limit, so does .
3. OK, so for question 1, the numerator grows quicker than the denominator so it goes to infinity?

As for 2), if I just take an even n, then it will just get bigger and bigger, so it doesn't have a limit. If the even powers don't have a limit, then the odd won't and so I can conclude that it doesn't have a limit? Am I right?
4. (Original post by DJB MASTER)
As for 2), if I just take an even n, then it will just get bigger and bigger, so it doesn't have a limit. If the even powers don't have a limit, then the odd won't and so I can conclude that it doesn't have a limit? Am I right?
Yes, quite. It's even more simple than that - the absolute values of the terms are unbounded.

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