# problem with infinite series equalling -1/12 ?

humy
Science 28 Oct '14 20:25
1. 28 Oct '14 20:256 edits
I have a problem with the infinite series:

S = 1 + 2 + 3 + 4 …

equalling -1/12

In one of the links I recently looked at, it said that the sum of the infinite series:

1 + ½ + ½ + ½ + ½ + ½ + ½ …

Is infinity. So surely the sum of the infinite series:

S1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 …

is also infinity?

OK, assuming this is the case, the infinite series:

S = 1 + 2 + 3 + 4 …

can be rewritten as:

1 + (1+1) + (1+2) + (1+3) + (1+4) …

which means it is equal to the sum of the two infinite series:

S1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 …

AND

S = 1 + 2 + 3 + 4 …

i.e. S = S1 + S

But note that S1 = infinity (I assume ) and the previous proof apparently proved S = -1/12. So surely, -1/12 + infinity equals infinity thus S is infinity?
(and this is not to mention the apparent logical contradiction of the values of the two sides of the equation being not equal )

I can only assume I am wrong but, what exactly is the flaw with my above logic?
2. DeepThought
28 Oct '14 22:00
Originally posted by humy
I have a problem with the infinite series:

S = 1 + 2 + 3 + 4 …

equalling -1/12

In one of the links I recently looked at, it said that the sum of the infinite series:

1 + ½ + ½ + ½ + ½ + ½ + ½ …

Is infinity. So surely the sum of the infinite series:

S1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 …

is also infinity?

OK, assuming this is the case, the infi ...[text shortened]... g not equal )

I can only assume I am wrong but, what exactly is the flaw with my above logic?
You have 1 + 1 + 1 + 1 + ... too big, S1 is definitely not positive:

Consider the difference -1/12 - (-1/12)
0 = -1/12 - (-1/12) = 1 + 2 + 3 + 4 + ... - (1 + 2 + 3 + 4 + ...)
= (1 - 0) + (2 - 1) + (3 - 2) + (4 - 3) + ....
= 1 + 1 + 1 + 1 + ...
So S1 = 1 + 1 + 1 + 1 +... = 0, which is just what one would expect of a field of characteristic zero.

Alternatively do the sum like this:

Let S = 1 + 1 + 1 + 1 + ...

and R = 1 - 1 + 1 - 1 + ...

S = 1 + 1 + 1 + 1 + ...
2S = 0+ 2 + 0 + 2 + ...
Therefore -S = S - 2S = 1 - 1 + 1 - 1 +... = R

R is given by the series 1 + x + x² + ... = 1/(1 - x) when x = -1, so R = 1/2.

Therefore 1 + 1 + 1 + 1 + ... = -1/2, which is negative so 1 + 1 + 1 + 1 + ... is definitely not positive infinity. I've found two ways it's non-positive, how many more do you want?
3. 28 Oct '14 23:12
Originally posted by humy
I have a problem with the infinite series:

S = 1 + 2 + 3 + 4 …

equalling -1/12

In one of the links I recently looked at, it said that the sum of the infinite series:

1 + ½ + ½ + ½ + ½ + ½ + ½ …

Is infinity. So surely the sum of the infinite series:

S1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 …

is also infinity?

OK, assuming this is the case, the infi ...[text shortened]... g not equal )

I can only assume I am wrong but, what exactly is the flaw with my above logic?
"I have a problem with the infinite series: S = 1 + 2 + 3 + 4 ... equalling -1/12"
--Humy

1 + 2 + 3 + 4 ... does not equal -1/12.

I don't know what else you may have read, and I would not even attempt
to guess about what you or someone else may misunderstand, but if your
conclusion is that 1+2+3+4... = -1/12, you should assume that someone
has done something incorrect at some point in arriving at that conclusion.
4. 28 Oct '14 23:233 edits
Originally posted by DeepThought
You have 1 + 1 + 1 + 1 + ... too big, S1 is definitely not positive:

Consider the difference -1/12 - (-1/12)
0 = -1/12 - (-1/12) = 1 + 2 + 3 + 4 + ... - (1 + 2 + 3 + 4 + ...)
= (1 - 0) + (2 - 1) + (3 - 2) + (4 - 3) + ....
= 1 + 1 + 1 + 1 + ...
So S1 = 1 + 1 + 1 + 1 +... = 0, which is just what one would expect of a field of characteristic zero.
...[text shortened]... nitely not positive infinity. I've found two ways it's non-positive, how many more do you want?
Thanks for that.

So does the infinite series

S1 = 1 + 1 + 1 + 1 +...

Actually equal zero?! (you said it can equal -1/2 in your second argument )

But I still have a confusion:

A link that I looked at said the infinite series:

1 + ½ + ½ + ½ + ½ + ½ + ½ …

equals infinity (see http://en.wikipedia.org/wiki/Harmonic_series_%28mathematics%29 under subtitle "Comparison test" )
But couldn't I rewrite that as:

1 + (½ + ½ ) + (½ + ½ ) + (½ + ½ ) …

= 1 + 1 + 1 + 1 +...

(is what I just did there in the above 'cheating'? )

= S1 ?
5. 28 Oct '14 23:345 edits
Originally posted by Duchess64
"I have a problem with the infinite series: S = 1 + 2 + 3 + 4 ... equalling -1/12"
--Humy

1 + 2 + 3 + 4 ... does not equal -1/12.

I don't know what else you may have read, and I would not even attempt
to guess about what you or someone else may misunderstand, but if your
conclusion is that 1+2+3+4... = -1/12, you should assume that someone
has done something incorrect at some point in arriving at that conclusion.
Actually, it is correct! Ridiculous as it may seem, it has been mathematically proven!

See for proof.
The said proof goes something like:

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 … = ?

S1 = 1 – 1 + 1 – 1 + 1 – 1 + 1 + 1 … = ½ (IF this sequence is infinite and not finite)

S2 = 1 – 2 + 3 – 4 + 5 – 6 + 7 … = ?

2S2 =
S2 + S2 =
1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 …
+ 1 – 2 + 3 – 4 + 5 – 6 + 7 …
= 1 – 1 + 1 – 1 + 1 – 1 + 1 + 1 …
(because each "+1" comes from 3+-2=1, 5+-4=1 etc and each "-1",comes from -2+1=-1, -4+3=-1 etc)
= S1 = ½

therefore, if 2S2 = ½, S2 = ¼.
So:

S2 = 1 – 2 + 3 – 4 + 5 – 6 + 7 … = ¼ (IF this sequence is infinite and not finite)

So, now:

S – S2 =
1 + 2 + 3 + 4 + 5 + 6 + 7 …
– [ 1 – 2 + 3 – 4 + 5 – 6 + 7 … ]
= 0 + 4 + 0 + 8 + 0 + 12 + 0 …

(note that, for example, the “+ 4 “ above comes from +2 – -2 = +2 + 2 = 4)
thus:

S – S2 = 4 + 8 + 12 + 16 + 20 …

factorizing 4 out gives:

S – S2 = 4( 1 + 2 + 3 + 4 + 5 + 6 + 7 … )

but note that the “1 + 2 + 3 + 4 + 5 + 6 + 7 …” part above is the wanted S thus

S – S2 = 4S

And, because S2 = ¼, means that:

S – ¼ = 4S

subtract both sides of the equation by S i.e. S – ¼ – S = 4S – S gives:

- ¼ = 3S

Divide both sides of the equation by 3 to get -1/12 = S i.e.

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 … -1/12 (IF this sequence is infinite and not finite)

But I am still confused by this result even though the above proof appears perfectly sound.
6. 29 Oct '14 00:042 edits
Originally posted by humy
Actually, it is correct! Ridiculous as it may seem, it has been mathematically proven!

The said proof goes something like:

S = 1 + 2 + 3 + 4 + 5 + 6 + 7 … = ?

S1 = 1 – 1 + 1 – 1 + 1 – 1 + 1 + 1 … = ½ (IF this sequence is infinite and not finite)

S2 = 1 – 2 + 3 – 4 + 5 – 6 + 7 … = ?

2S2 ...[text shortened]... not finite)

But I am still confused by this result even though the above proof appears sound.
A recent thread was created by Shavixmir, who apparently struggled to
comprehend that the convergent infinite series 1/2 + 1/4 + 1/8 ... = 1.
I don't think changing the subject to divergent infinite series is helpful.

As a general rule, I don't watch YouTube videos. Believe it or not, I
learned mathematics from studying books, articles, and solving problems.
Can you refer me to a book by a reputable mathematician with your claim
in it? If I have the time to read it, then I shall consider that argument.

It's a divergent series. I know that a divergent infinite series such as
1 - 1 + 1 - 1 ... may be *assigned* the value of 1/2 by a summation method
(averaging) such as a Cesaro summation. I would submit that it would be
misleading at best to write as though a divergent series *must equal*
something as a sum in any classical sense, when the result should depend
upon the summation method chosen. Insofar as I am aware, there are
many summation methods, which could assign different values to the same
divergent series.

Summation methods were not an area of my former mathematical expertise.
As a child, however, I intuitively quickly understood that the harmonic
series is divergent, and I came up with a simple proof for it.
7. 29 Oct '14 00:272 edits
Originally posted by Duchess64
A recent thread was created by Shavixmir, who apparently struggled to
comprehend that the convergent infinite series 1/2 + 1/4 + 1/8 ... = 1.
I don't think changing the subject to divergent infinite series is helpful.

As a general rule, I don't watch YouTube videos. Believe it or not, I
learned mathematics from studying books, articles, and solving ...[text shortened]... kly understood that the harmonic
series is divergent, and I came up with a simple proof for it.
I found this:
http://en.wikipedia.org/wiki/Grandi%27s_series
not yet sure what I should make of it.
8. DeepThought
29 Oct '14 18:59
Originally posted by humy
I found this:
http://en.wikipedia.org/wiki/Grandi%27s_series
not yet sure what I should make of it.
You should read the page "Divergent series" which explains the issues quite well and gives a set of techniques for assigning values to series which aren't absolutely convergent. It's also worth having a look at the page "Riemann series theorem" which shows that the sum of a series which isn't absolutely convergent can be summed to anything you want it to be summed to.

http://en.wikipedia.org/wiki/Riemann_series_theorem
http://en.wikipedia.org/wiki/Divergent_series
9. 29 Oct '14 20:31
Originally posted by DeepThought
You should read the page "Divergent series" which explains the issues quite well and gives a set of techniques for assigning values to series which aren't absolutely convergent. It's also worth having a look at the page "Riemann series theorem" which shows that the sum of a series which isn't absolutely convergent can be summed to anything you want it t ...[text shortened]... tp://en.wikipedia.org/wiki/Riemann_series_theorem
http://en.wikipedia.org/wiki/Divergent_series
Thanks. Yet another thing I will now study.
10. 29 Oct '14 21:23
Originally posted by humy to DeepThought
Thanks. Yet another thing I will now study.
At a local university, every student who seeks an undergraduate degree in
mathematics is required to pass a basic course about mathematical proofs.
Given that most of these students will become teachers, it's important that
they have enough training to distinguish, at least at a basic level, their
future students' valid proofs from their invalid proofs.

As far as I can infer, you (Humy) never have been exposed to such a course
mathematics, I would submit that you might well be misled if you keep
attempting to learn rather advanced mathematics from isolated Wikipedia
articles or YouTube videos. University students don't use Wikipedia and
YouTube as their primary resources for textbooks and lectures. Wikipedia
articles don't test students' understanding by making them solve problems.
Mathematics cannot be learned just by reading; one must solve problems too.

There's not a quick and easy way to learn much advanced mathematics.
Even for me (I had extraordinary natural gifts), I would have to work at it.
So if you are really seriously interested in developing a sound understanding
of mathematics, then I would advise you to find a textbook that suits your
interests and work your way through it from the beginning.
11. 29 Oct '14 21:495 edits
Originally posted by Duchess64
At a local university, every student who seeks an undergraduate degree in
mathematics is required to pass a basic course about mathematical proofs.
Given that most of these students will become teachers, it's important that
they have enough training to distinguish, at least at a basic level, their
future students' valid proofs from their invalid proofs ...[text shortened]... u to find a textbook that suits your
interests and work your way through it from the beginning.
I used to know all this stuff a very long time ago when I was much younger because I did university maths courses. Unfortunately, it is just terrible how much of it I forgot since then and now have to relearn. Although, having said that, I am finding that, due to the recent return of interest in maths, my memory is sometimes jogged by what I read on the web and then much of that old knowledge comes flooding back to me and I think "arr yes! that's right! now I remember how to do all that! "

The reason for my sudden recent interest in maths is because I am researching a highly original theory (about probability and how to completely solve the problem of induction! ) I independently came up with and am now working on that I thought unfortunately requires some very difficult mathematics. But, fortunately, I have just worked out the other day that I don't need such complex maths for this after all and I can get away perfectly fine with much simpler basic maths with the only type of proof needed being proof by contradiction which I don't have to look up because I remember how to do that perfectly and I generally find proof by contradiction a very easy (and satisfying ) type of proof to construct.
12. 29 Oct '14 22:14
Originally posted by humy
I used to know all this stuff a very long time ago when I was much younger because I did university maths courses. Unfortunately, it is just terrible how much of it I forgot since then and now have to relearn. Although, having said that, I am finding that, due to the recent return of interest in maths, my memory is sometimes jogged by what I read on the web and ...[text shortened]... generally find proof by contradiction a very easy (and satisfying ) type of proof to construct.
that your mathematical background was very modest at best.
13. 29 Oct '14 23:15
Originally posted by humy
I have a problem with the infinite series:

I can only assume I am wrong but, what exactly is the flaw with my above logic?
Your logic is flawless. You definitely have a problem with infinite series.
14. 30 Oct '14 05:54
Originally posted by Duchess64
I would submit that you might well be misled if you keep
attempting to learn rather advanced mathematics from isolated Wikipedia
Given that you have already admitted that you don't watch YouTube videos, I find this advice somewhat suspect. Also, given that you clearly went to school prior to the advent of online video, your personal experience is hardly relevant.
YouTube videos and Wikipedia are excellent teaching aids. If schools are not yet using them, then they should be. And despite your claims, it is actually possible to do university courses, including ones in mathematics, that have YouTube as the primary medium of instruction.

https://www.edx.org/course-search?search_query=mathematics
15. 30 Oct '14 08:412 edits