Thread: 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 =

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

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?

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

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.

Another way (using the definition of limit):



This condition is satisfied for

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.