Re: Another Sequence Problem

Re: Another Sequence Problem

Quote:

Originally Posted by

**earboth** I'll do the first re-arrangements to get you started ...

When calculating the limit keep in mind that

The final result is

Thanks for the help and feedback, I really appreciate it. This is the only problem out of the sequences homework that I really couldn't understand. Actually, I am still not quite sure how you came up with your solution. Could you expand on the method more or tell me where I could find out more about it?

Thanks

Re: Another Sequence Problem

Re: Another Sequence Problem

I really have no idea, about this convergence topic, but... im trying to understand your solution step by step.... hmmm sir, may I ask, how did you use synthetic division to make (n/(n-8))^5n into (1 -(8/n-8))^5n? I really appreciate if you could explain it to me... hmmm oh yeah.. how did you use the concept of property of fraction to make 8/n-8 to 8/n? thank you very much...

Re: Another Sequence Problem

Quote:

Originally Posted by

**earboth** I don't know from your reply which part of my answer isn't understandable for you. So here is the complete version:

...... Both powers have the same exponent so they could be collected to a single base.

By the "property of a fraction" I mean: If the denominator increases the complete fraction decreases. That explains the ">"-sign.

Since

this explains the equal relation in

when you are going to determine the limit.

Now calculate the limit:

If

approaches

then

approaches

too.

Therefore

. Replace n with 8t and keep in mind that

Wow! Thanks for the detailed steps for this problem. I am still puzzled as to what you did, for example the synthetic division and property of fractions. I don't ever remember learning this particular method. Is it possible to use L'hospital's rule? For example, this problem is also the same as (n/n-8)^5n, so couldn't you use the natural logarithm and then l'hospitals rule?

Re: Another Sequence Problem

Quote:

Originally Posted by

**kspkido** I really have no idea, about this convergence topic, but... im trying to understand your solution step by step.... hmmm sir, may I ask, how did you use synthetic division to make (n/(n-8))^5n into (1 -(8/n-8))^5n? I really appreciate if you could explain it to me...

There are several possible ways to do the transformation:

or Code:

` 8`

n ÷ (n - 8) = 1 + -----

-(n - 8) n - 8

--------

8

This is synthetic division as it is taught in Germany.

Quote:

hmmm oh yeah.. how did you use the concept of property of fraction to make 8/n-8 to 8/n? thank you very much...

You probably have noticed that . So when I changed the denominator to n **the denominator was enlarged** by 8. Thus the value of the whole **fraction became smaller**. That's all.

Re: Another Sequence Problem

Quote:

Originally Posted by

**Beevo** Wow! Thanks for the detailed steps for this problem. I am still puzzled as to what you did, for example the synthetic division and property of fractions. I don't ever remember learning this particular method. Is it possible to use L'hospital's rule? For example, this problem is also the same as (n/n-8)^5n, so couldn't you use the natural logarithm and then l'hospitals rule?

This is basic to everything done here:

That can be generalized to:

So here is an easy example:

Re: Another Sequence Problem

Got it... I'll try and memorize this method. Thanks

Re: Another Sequence Problem

Quote:

Originally Posted by

**Beevo** This question asks whether the problem is converging (find limit) or diverging.

an= (n)^5n/(n-8)^5n

I first took the natural log of both the numerator and denominator and got ln(n)/ln(n-8), and then used l'hospitals rule to get a complicated mess. Can anyone give me some input on what went wrong, or if I just started out in the incorrect manner.

Thanks

An easier method:

Re: Another Sequence Problem

Quote:

Originally Posted by

**Prove It** An easier method:

Thanks for this detailed response.