To whom may I concern:
Hello. I am finian. I visit your site quite often. So sorry for my awful english.
Just go back to business. I encounter an question in a book.I can't due with it. The question is as follows:

¡Û
(sigma) (((n+1)/(n+3))^n)
n=1

(Sorry, I don't know how to type the sigma notation) Does this series converge or diverge? How to prove for convergence or divergence?
I asked my teacher, he told me to post at here.Could anyone please help me to solve it?
Sorry to bother you.I am looking forward to your reply.
Yours
Finian
A secondary 4 Student

> I asked my teacher,
>he told me to post
>at here.

You have a very clever teacher, indeed. May he/she also send a small part of his/her salary my way?

A general term of any convergent series tends to zero. Does this hold true for your series?

Dear Alexb:
But the thing is:
HOW TO FIND : limit (general term)?
n-infinity

I can't think of any rules & test for limits that can find it. Can you help me, please?
Finian

> I
>can't think of any rules
>& test for limits that
>can find it. Can you
>help me, please?

I am sorry. I may only help you if you try helping yourself. Tell your teacher he/she sent you to a wrong guy.

You may try posting your question to the sci.math newsgroup. There's always somebody willing to solve somebody's homework. Do not know if you should call this help. Help is helping to study, not just supplying solutions.

If you still want me to help, then open your textbook, find limits that look like the general term of your series, make at least one step - try something. If you fail, I may be able to help you with the next step.

Alexb:
The question is not a homework. I do it for my own intrest.
You think that the question is in the syllabus of S-4?
With a calculator and subsituting big values, it can be shown that the series diverges.I knew this before I post the question. But how can I do it with mathematical means?
I tried hard to find the limit of the general term, but the term is ¡u¡Û/¡Û¡v,so I diffentiate it. But logarithmetic diffrentiation keeps on giving back exponential invloves nth or composition of nth power.So ratio tests and the n-th term test for divergence is of no use.
I have think of comparison with e, because after some algebaric operations, the term is something like (1+(something)/(f(n)))^n ,like e very much. But I can't do further to make the term be something related with e.
I've done anything I can do,including ask anyone who may know how to do it.So, can you help me?
Finian

LAST EDITED ON Apr-11-01 AT 09:03 AM (EST)

LAST EDITED ON Apr-11-01 AT 08:46 AM (EST)

The problem is I can't start from ground 0. I do not know your course' syllabus, but I am pretty sure that a decent textbook with a title "Calculus" on the cover has some pertinent material.

Look for the limit as n goes to infinity of (1+1/n)n. As you like the number e, you'll be happy to learn that this is the limit of the sequence.

After that, proceed in steps:

lim (1+a/n)n = ea, a is constant.
lim ((n+1)/(n+3))n = lim ((1 - 2/(n+3))n+3)n/(n+3)

Alexb:
Sorry, but I really can't see the words after the sentence'Look for the limit as n goes to'.
You can think that I've done the basic course of dif. and int., but series is something new to me.I knew a little about limits.I also have a 'decent' calculus book.
Finian

Alex:
Really thanks for the three reply.You really helped me to cut these three knots!Thanks a lot!
Finian