problem with infinite series equalling -1/12 ?

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?

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?

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 ?

The post that was quoted here has been removed

Actually, it is correct! Ridiculous as it may seem, it has been mathematically proven!

See YouTube 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.

The post that was quoted here has been removed

I found this:
http://en.wikipedia.org/wiki/Grandi%27s_series
not yet sure what I should make of it.

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

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.

The post that was quoted here has been removed

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.

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.

The post that was quoted here has been removed

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

Originally posted by twhitehead
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 d ...[text shortened]... the primary medium of instruction.

https://www.edx.org/course-search?search_query=mathematics

https://www.edx.org/course-search?search_query=mathematics

Thanks for that! I know you can do open university courses via the net but I think I look into these courses as well.

I just don't understand why [some] people here are implying you cannot learn maths over the net! WHY not? That doesn't make sense.
And what is so terrible about YouTube videos and Wikipedia anyway? Sure there are some YouTube videos that are full of a load of crap (mainly Creationists propaganda videos ), but, it is easy enough to just ignore the ones that are obviously full of crap and is there a mysterious logical contradiction ( that I am unaware of ) in a YouTube video or a Wikipedia page containing 100% perfectly valid information and therefore allow one to learn well from it i.e. be perfectly educational?