Re: Need Help on these 2 problems.

To: CALC-TI@LISTS.PPP.TI.COM
Subject: Re: Need Help on these 2 problems.
From: RWW Taylor <RWTNTS@RITVAX.ISC.RIT.EDU>
Date: Mon, 1 Jun 1998 10:23:28 -0400
Reply-To: RWW Taylor <RWTNTS@RITVAX.ISC.RIT.EDU>

Garvin Cung writes:

> does anyone here know how to prove these 2 questions? :
>
> Q1)
> 1 + 1/2 + + 1/3 + 1/4 + .....
> prove how is this a divergent(no limit) series
>
> Q2)
> 1 + 1/4 + 1/9 + 1/16 +.....
> prove how this is a convergent(there is a limit) series
>
> reply as soon as possible. thanks a lot!

You don't say where you are coming from on these problems, what tools you want
to allow in your proof, etc.

The standard proof of the divergence of the harmonic series (your first
problem) is very simple -- note that 1/3 + 1/4 is greater than 1/2, and
1/5 + 1/6 + 1/7 + 1/8 is greater than 1/2, and the sum of the next eight terms
is greater than 1/2, and so on.  Another neat way to group these terms to see
the divergence is given in the current issue (Spring 1988) of the journal
Mathematics and Computer Education.

The second series is not so susceptible to elementary analysis, though I
believe I have seen a clever diagram offering a "Proof without Words". Both
series are examples of the general series sum(1/n^p), which is convergent for
all numbers p>1. The so-called "integral test" is one good way to show this,
if you have access to elementary notions of calculus.

RWW Taylor
National Technical Institute for the Deaf
Rochester Institute of Technology
Rochester NY 14623

>>>> The plural of mongoose begins with p. <<<<