Liz F

Posted on Monday, 27 January, 2003 - 05:38 pm:

I'm working through the MEI P6 chapter on limits and I'm stuck on a question. Any hints would be very helpful.

The alternating series u1 + u2 + u3 +... is such that

(A)un tends to 0, as n tends to infinity
(B) |un|> |un+1|

Assume that u1 is positive. Using the usual notation for partial sums prove that :

(i)S1,S3,S5,... form a decreasing sequence which is bounded below by 0.
(ii) S2,S4,S6,... form an increasing sequence which is bounded above by u1.
(iii) Both sequences tend to the same limit and deduce that u1+u2+u3+... converges.

It seems that (for (i) and (ii) to be true) I need to prove that all the even terms of the sequence are positive and all the odd terms are negative, but I don't really see how to do this. Also, it seems obvious that both sequences tend to the same limit (since each term of one sequence is a term from the other with a un added on, which tends to 0 as n tends to infinity). Is there some way of writing a formal proof of this?

I can't find anyway of writing in subscript, everything which comes after S or u should be in subscript - sorry!

Thanks
Liz

Demetres Christofides

Posted on Monday, 27 January, 2003 - 06:57 pm:

The series is alternating. What does this mean ?

Demetres

Vicky Neale

Posted on Monday, 27 January, 2003 - 07:14 pm:

Just a comment - for tips on how to do formatting (such as subscripts), have a look at the Formatting button at the bottom of the page. You'll find superscripts and subscripts there, as well as a load of other stuff.

Vicky

Steve Megson

Posted on Monday, 27 January, 2003 - 07:21 pm:

The -{ } tag produces subscript, for example S-{1} produces S₁ . See the formatting help for more.

Steve

Liz

Posted on Monday, 27 January, 2003 - 07:43 pm:

Thanks! I just looked up "alternating": I was under the impression it just meant the same as oscillating.

Steve Megson

Posted on Monday, 27 January, 2003 - 08:38 pm:

You can prove (iii) from the definition of convergence. It's basically your argument with some

ε

s added.

$S_{n} \to S$ if for any $ε > 0$ there exists an $N$ such that for any $n > N$ , $| S_{n} - S | < ε$ .

Assume that $S_{1}$ , $S_{3}$ , $S_{5}$ , ... converges to $S$ , and take any $ε > 0$ . There is an $N$ such that for any odd $n > N$ , $| S_{n} - S | < ε / 2$ and $| u_{n + 1} | < ε / 2$ .

Then $S_{n + 1} = S_{n} + u_{n + 1}$ is within $ε$ of $S$ . Hence $S_{2}$ , $S_{4}$ , $S_{6}$ , ... tends to $S$ as required.

Steve