Convergence of a sequence example

Starting to look at Convergence of sequences and given an example without a solution. The solution would really help my understanding of what's going on here:

Define what is meant by saying converges to the limit l as n goes to infinity. Show that converges to as n tends to infinity.

Now I know you can pull out the fraction and show that the rest tends to 1 (right?), simple enough you can already see the right there but how do you show this? Basically that l =

Re: Convergence of a sequence example

Perhaps you got typo, as written, the sequence diverges.

Re: Convergence of a sequence example

Quote:

Originally Posted by

**Shizaru** Starting to look at Convergence of sequences and given an example without a solution. The solution would really help my understanding of what's going on here:

Define what is meant by saying

converges to the limit l as n goes to infinity. Show that

converges to

as n tends to infinity.

There is a real problem with your post:

If it were that is correct.

Which is it?

Re: Convergence of a sequence example

Yep, sorry, both powers are 3. Corrected it.

Re: Convergence of a sequence example

Quote:

Originally Posted by

**Shizaru** Starting to look at Convergence of sequences and given an example without a solution. The solution would really help my understanding of what's going on here:

Define what is meant by saying

converges to the limit l as n goes to infinity. Show that

converges to

as n tends to infinity.

Now I know you can pull out the fraction and show that the rest tends to 1 (right?), simple enough you can already see the

right there but how do you show this? Basically that l =

The standard way to deal with a problem like this is to divide top and bottom of the fraction by the highest power of n in sight, and then use some theorems about limits of sums, products and quotients. In this case, the first step is to divide through by :

In the numerator, as There is a theorem saying that the limit of a sum is the sum of the limits. That theorem tells you that the limit of the numerator is 9+0=9. Similarly, the limit of the denominator is 2+0=2. Finally, another theorem says that if the numerator and denominator of a fraction both have limits (and the denominator has a nonzero limit) then the limit of the quotient is the quotient of the limits. Apply that theorem to see that the limit of is 9/2.

Re: Convergence of a sequence example

Another way (using the definition of limit):

This condition is satisfied for

Re: Convergence of a sequence example

Thanks - how would I write up something like this. I know the theorems, and that basically each part can be taken on its own, like as you put it sum of limits = limit of sum and likewise for quotient. I know these rules but I don't know how to express it... to me it just seems like proving 1 + 1 = 2, its intuitive but I need a way of expressing it... I hope you get what I mean,,

Plus what I was trying to say is that I would rearrange into ( )( ) show that the second bracket tends to 1 .. is this the same as what you were saying? (just the opposite?) this was my initial working.