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veeceeone

Member

Registered: 2019-10-29

Posts: 8

The Last Few Terms in an Infinite Series

I don't understand "infinity" at all but can live with it. [ What choice do I have? :-) ]

I do understand that the last few terms have extraordinarily small values in a series like:

S = 1/2 + 1/4 + 1/8 + 1/16 + ...

The following is from the proof for the sum of this series, which I think is outstanding. My question is a conceptual one.

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First, we will call the whole sum "S":   S = 1/2 + 1/4 + 1/8 + 1/16 + ...
Next, divide S by 2:S/2 = 1/4 + 1/8 + 1/16 + 1/32 + ...
***

Let's say that S has n terms. (I understand that n is  ∞.)
S/2 may be thought of the terms in S shifted to the left by one position throwing out the first term. This series also will have n terms. What is the value of the nth term? This term will not have a corresponding term in S.

Am I just wasting my time asking questions like this? The proof is great by the way. I can now find the sum of geometric series like

1/8 + 1/64 + 1/512 etc very easily. [I didn't do the math. I am guessing that it is 1/7.]

veeceeone

Member

Registered: 2019-10-29

Posts: 8

Re: The Last Few Terms in an Infinite Series

Summing geometric infinite series with ratios of 1/2, 1/3, 1/4, etc.

1. Divide by the denominator to get a new infinite series.

2. Subtract the second series from the first series. This will get rid of all the infinity mess and will get results like

S/2 = 1/2  for a ratio of 1/2. S = 1

2/3 S = 1/3  for a ratio of 1/3. S = 1/2

3/4 S = 1/4  for a ratio of 1/4. S = 1/3

4/5 S = 1/5  for a ratio of 1/5. S = 1/4

5/6 S = 1/6  for a ratio of 1/6. S = 1/5

6/7 S = 1/7  for a ratio of 1/7. S = 1/6

7/8 S = 1/8  for a ratio of 1/8. S = 1/7

One can generalize:

(n-1)/n S = 1/n for a ratio of 1/n. S = 1/(n-1)

Any typos aside, am I right? I realized all these only after seeing the proof in MathIsFun. I am not showing off. This may be of interest to others too. However, if you want me to stop posting posts like this, let me know and I will stop.

Whether I post or not, this is the kind of activity that your site evokes in me. Posting motivates me to complete this process.

Thanks.

veeceeone

Member

Registered: 2019-10-29

Posts: 8

Re: The Last Few Terms in an Infinite Series

I also realized this.

The sum of a geometric infinite series with a ratio < 1 can also be calculated as follows:

1. Multiply by the denominator of the ratio to get another infinite series.

2. Subtract the original series from this.

For S = 1/2 + 1/4 + 1/8 + 1/16 +......

2S = 1+  1/2 + 1/4 + 1/8 + 1/16 +......

2S-S = 1
S=1

The more general solution is this (after multiplying by n and subtracting the original series from the new series; 1/n is the ratio; n > 1).

S(n-1) = 1
S = 1/(n-1)

Last edited by veeceeone (2019-11-02 01:03:03)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: The Last Few Terms in an Infinite Series

hi veeceeone

I doubt that anyone can 'understand' infinity.  You cannot see it or be aware of it with any other sense.  But it does have its uses.

Before I start on that a word about mathematics.  Maths doesn't exist in the above sense either … but you can use it for modelling.  There's a whole load of theory used to develop arithmetic for example.  This theory works on paper and could just be a study in its own right.  But most people use it when the model can be applied such as working out your bank balance; counting your sheep; deciding if you've bought enough marshmallows for everyone at your party to have three each.

But there are also situations where the rules of arithmetic don't work: eg.  when you add one pile of sand to another pile of sand the result is one pile of sand (1+1=1). eg.  show me half a piece of chalk.

You might have learnt lots of Euclidean geometry, but take care trying to use Pythagoras' theorem on the surface of a sphere.

So there's a branch of maths that considers what happens if you repeat something for ever.  Obviously we cannot really do this but we can ask what if?

There's a story about a frog that can jump 1/2 a metre on the first jump, then 1/4 m then 1/8 m and so on.  If it carries on jumping for ever how far will it jump altogether?

Put a line to indicate the frog's starting point and another exactly 1 metre away.  The frog covers half the distance on it's first jump; then half what is left on the second; then half of what's left on the third jump and so on.  It will get closer and closer to the 1 metre line as time goes by.  Will it reach the line?  Well no, because there's always a tiny bit left to cover.   But real frogs cannot jump so precisely and the line we made will have some thickness so eventually you won't be able to tell that the frog hasn't arrived on the line. So it is reasonable in practice to ignore the tiny distance left, and accept that the frog does get there in the real world.  In mathematics we make it sound more rigorous by saying that the limit as n tends to infinity of the series 0.5 + 0.25 + 0.125 + ……. is 1.

There's as much validity in that as assuming that lines have no thickness or that you can actually measure the three angles of a triangle, add them up and get 180.  I've tried this with a class and we got answers ranging from 178 to 182.

Now to the sums.

Let's say that S has n terms. (I understand that n is  ∞.)

In the theory of infinite sums I afraid the above is wrong on two counts.  The series has no last term.  And mathematicians don't allow infinity to be a number.

So, when you subtract one series from another the subtraction process goes on for ever.  So there is no last term left over.  The parts of the two series that go on for ever 'cancel' each other out completely and so you're left with some algebra you can deal manage without worrying about infinity.

If you look here:https://www.mathsisfun.com/algebra/sequ … etric.html

you'll find the same trick used to find the sum of n terms of a geometric series and then the sum to infinity of such a series.

Your investigative result is correct (well done!). It is a special case of the above sum formula with a = 1

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

veeceeone

Member

Registered: 2019-10-29

Posts: 8

Re: The Last Few Terms in an Infinite Series

Hi Bob,

Thanks for your reply. It gives some something to think about and work on.

So, when you subtract one series from another the subtraction process goes on for ever.  So there is no last term left over.  The parts of the two series that go on for ever 'cancel' each other out completely and so you're left with some algebra you can deal manage without worrying about infinity.

I understand and agree. This is a neat trick and this is what excited me. However, if you look at just one infinite series, there is a lot of hand waving going on.

I doubt that anyone can 'understand' infinity.

How can people discuss this with each other if they cannot understand it? Also, when someone proposed it first, how could the others agree that something well-defined was being proposed?

Your investigative result is correct (well done!)  It is a special case of the above sum formula with a = 1.

Thanks. Doing stuff like this is a part of the fun for me of being in sites like this. Thanks for your link. I will look at it.
VC1

Last edited by veeceeone (2019-11-02 05:43:21)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,619

Re: The Last Few Terms in an Infinite Series

hi veeceeone

Oh great!  It will be such fun having you as a member.

How can people discuss this with each other if they cannot understand it?

But what have we been doing?

As for the second part; the 'validity' of any theory is tested by "does it work"?

Richard Feynman said: "If you think you understand quantum mechanics, you don't understand quantum mechanics."

And, slightly off the topic of understanding but even more fun,  from Lewis Carroll (real name Charles Dodgson … he was a mathematician as well as writing Alice in Wonderland):

"Alice laughed: "There's no use trying," she said; "one can't believe impossible things."
"I daresay you haven't had much practice," said the Queen. "When I was younger, I always did it for half an hour a day. Why, sometimes I've believed as many as six impossible things before breakfast."

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

veeceeone

Member

Registered: 2019-10-29

Posts: 8

Re: The Last Few Terms in an Infinite Series

Richard Feynman said: "If you think you understand quantum mechanics, you don't understand quantum mechanics."

I have heard that. If nobody understands it, how can they work with it? Thanks for your input. I will go on to enjoying your site.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: The Last Few Terms in an Infinite Series

Old post 0.999... = 1?
http://www.mathisfunforum.com/viewtopic.php?id=658

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X'(y-Xβ)=0