Convergence and divergence activities in Sofia

Examples of decreasing sequences

We asked the kids to give some examples of decreasing sequences.

Here are the first examples by Teddy and Mitty (13 years old, in a specialised mathematics school).

Teddy: a-1, a-2, a-3,...

Mitty: 0, -1, -2, -3,...

 Then we introduced a restriction for the sequences to contain only positive numbers.

They came up with the following:

Teddy: a², (a-1)², (a-2)², ...

Mitty: x⁰, y⁰, z⁰, ...

And they realised that there was a mistake – the first sequence was not decreasing “all the time”, and the second was “non-increasing” rather than “strictly decreasing”.

Teddy’s next approximation was:

aⁿ, a^n-1, ...

 a=5, n=10

At the point when the power became a negative number Teddy stopped for a while and said that they hadn’t studied negative powers yet but that he had noticed this notation in the physics lab of their school and that 10⁻¹ was in fact 0.1. Then he checked the behaviour of the sequence aⁿ for a>1 and a<1 by writing down the first members of the sequence in both cases.

Mitty’s example was more concrete:

(1/2)², (1/4)², (1/16)²,...

 And he realized that he could simplify his example by getting rid of the powers.

Finally we came to the idea of generating the sequences {1/n} and {1/n²}.

Which sequence is easier to generate?

Before they turned the computers on we encouraged them to choose a sequence to start implementing in TT and asked them which sequence was easier.

Mitty: Teddy’s sequence {1/n}. Since there is one operation fewer in his sequence — (raising to a power). No, wait.

Teddy: Please, wait, don’t shoot yet.

Mitty: In fact, my sequence {1/2ⁿ} is easier since every member can be obtained from the previous by dividing by … ½, I mean — by dividing by 2.

Teddy –Yes, his sequence is easier.

They started building their robots.

 After a while:

Mitty: You know, it depends on which key is closer to you – to divide by 2, or …

Teddy: Or first to add 1 and then – to divide 1 by this number.

Mitty: (looking at the numbers his robot generates). Ha, look how long my numbers are getting.

Jenny: Does this mean they are “big”?

Mitty: Not big at all – these are a million times, a billion times smaller than 1 since they are obtained after halving 1 many times.

The first version of Teddy’s ADD_UP robot was without using ADD1. After we encouraged him to use ADD1 he constructed a new version of ADD_UP whose output were the partial sums of 1/n in decimal representation. George taught him how to convert the numbers in fractions and he made a new iteration “with the special 1” as he called the 1 on a pad with fraction representation.

Jenny (to Teddy): One more time – which sequence was easier to build?

Teddy: Mitty’s sequence {1/2ⁿ} is easier. Pauses in reflection. Wait, maybe both are equally easy … if we have different versions of ADD1 (generating the denominators) – one – counting, and the other -doubling the numbers.

In another session Ivan (12 years old) first built the geometric progression and then – the harmonic series. He used decimal representation and the teacher did not interfere so as not to shift the focus of attention.

Ivan (to the same question): Maybe the first sequence, 1/2ⁿ, is easier since the only thing you have to do is to divide each member by 2. For the second sequence I used a robot with a 3-hole box – in the first one I put 1, in the second hole the robot should add 1 each time starting with 1, and I put a bird in the third hole. I trained the robot to divide the 1 from the first hole by the number in the second and to give it to the bird. I made a mistake the first time by copying the resulting number in the first box and giving it to the bird. Thus at the third step the robot divided 0.5 (instead of 1) by 3. But it was easy to retrain it by giving the resulting number to the bird (instead of copying it).

Can the partial sums of the harmonic series exceed 2?

In the next session Mitty started working on the questions related to the behaviour of the harmonic series (http://www.weblabs.org.uk/wlplone/Members/ioe/my_reports/Report.2004-03-03.3328/index_html)

 Mitty: Every member of the sequence 1+1/2+…+1/n …is bigger than the previous one but with a number which become smaller and smaller (as n grows).

(He reflected for a while and made a couple of calculations with his pocket calculator.

Surprisingly he didn’t immediately figure out that the difference between the third and the second partial sum of the harmonic series was just 1/3 and started calculating 1+1/2+1/3 and then subtracted 3/2 from it.)

My first feeling is that no matter how small the added number is the sum can exceed 2 because we add as many members as we wish.

Jenny: I don’t know. This is what we want to check.

Mitty: But you know, this is different from Zeno’s paradox.

Jenny: In what way?

Mitty: The turtle keeps doing the same thing – it always moves by a very small stretch, whereas Achilles… (thinks hard again)

Jenny: But there is a certain similarity, right? You add endlessly some numbers (the distances Achilles covers each time) which get smaller and smaller. The question is whether the covered distance will become larger than a concrete number?

Mitty: O.K. Let me build first the robot that generates 1/n and than the ADD_UP robot which will accumulate these numbers.

Till the end of the session Mitty was building the robots and was very disappointed that he had to leave before he could check if the ADD_UP would generate a number bigger than 2. He asked for permission to come again the next day so as to check this.

This time he rebuilt the robots much faster on Jenny’s computer while she was examining in-service teachers.

Jenny: Is it strange that the partial sums of the harmonic series became bigger than 2?

Mitty: Yes and no. Because the further we go the smaller the added number is. On one hand we can go as far as a billion members of 1/n and the feeling is that we can go over 2, but on the other we add hellishly small numbers and it is not clear if it will reach 2.

Ivan (during another session): I think that both sequences would not exceed 2.

First he constructed the robot for generating the sequence {1/n}:

His ADD_UP robot was with a 2-hole box. In the first he had put the nest of the robot, generating the sequence {1/n}, and the second hole was for accumulating the consecutive members of this sequence. This version turned out to be the tidiest one for watching the relation between n and 1+1/2+…1/n.

He started watching the robot generating the harmonic series: Ha, my robot doesn’t seem to work properly!

Jenny: Why?

Ivan: Because it already generates numbers bigger than 2!

Jenny: Maybe your conjecture was not right…

Ivan: Maybe…

He constrcuts the robot for the geometric progression with a quotion 1/2 (the first version was without a bird since he is a fan of the shortest possible constructions and then he added a bird on oour request) :

After watching the robot generating the geometric progression:

And these numbers get closer to 2 never seeming to exceed it …

I showed the kids in this session the formal (relatively very simple) proof of the divergence of the harmonic series and the fact that if we know that the geometric series is convergent its sum should be 2 http://www.weblabs.org.uk/wlplone/Members/Sofia/my_reports/Report.2004-06-24.3055. Ivan seemed very happy but Yana appeared irritated. She started writing on the blackboard:

Yana: How can it be: on one hand S (the sum 1+1/2+1/2²+…) has an infinite number of terms so it should be infinity, on the other – it is 2. As for Zeno’s paradox there is a contradiction in the formulation. It is not possible for Achilles to be faster than the turtle if he can’t overtake it!