What is the sum of this infinite series?

[jonjacson]

Sum= ...- 1 + 1 -1 +1-1+1... until infinite

It is just an infinite sum of -1 plus 1.

Can anyone tell me the sum of this infinite series and a demonstration of that result?

THanks!

 

[hilbert2]

That series is not convergent in the traditional sense, so strictly speaking it does not have a sum. However, the series is Cesaro summable and has sum 1/2 in that sense. See http://en.wikipedia.org/wiki/Cesàro_summation .

 

[jonjacson]

So the answer is that there is not a definite answer, in case we can associate a number we could pick the 1/2 number. Is that right?

I wonder what I am doing wrong with this reasoning:

1.- I think the answer is 0, since we can arrange the infinite series in pairs, for every -1 there will always be a 1, (-1 + 1) +(1-1) etc. As every one of those pairs sums 0, the total sum of the infinite seres should be 0. What is wrong? Why is it better 1/2 than 0?

 

[hilbert2]

The series can not be summed in the usual sense. Its partial sums don't approach any particular value, they oscillate between 0 and 1. However, the mean of first n partial sums approaches 1/2 when n is given larger and larger values, so we can assign the series a Cesaro sum of 1/2.

 

[jonjacson]

My partial sums are equal to zero always, I just need to add two numbers to the series to get more and more 0, and I always get 0 as a result. What am I doing wrong?

 

[hilbert2]

Before you can calculate sums of infinite series, you need to define accurately what a "sum" means in the case there are infinitely many terms. I suggest you first make yourself familiar with the epsilon-delta definition of a limit, to see how things are defined in mathematics. If you don't have a strict definition, you can assign that sum ANY integer value by just rearranging terms.

 

[jonjacson]

But the way I used to rearrange terms includes tem all.

I have seen somewhere that they rearranged the terms and said the sum was -1 :

-basically they said -1 (+1 -1)+ (+1-1) etc should be equal to -1 + 0 +0 +0 ... so the sum was -1.

-and viceversa it could be the same with +1

But if you "isolate" +1 I think somewhere else there must be a pair for him which is a -1
If you isolate -1, there must be a +1 somewhere that is his couple

But using my reasoning we are not forgeting any number, it looks fair to me, we just get together all the numbers in pairs, since every sum is 0, the total sum must be 0. It looks so clean that I cannot see where is the fallacy, or the mistake with that reasoning.

 

[Mark44]

My partial sums are equal to zero always, I just need to add two numbers to the series to get more and more 0, and I always get 0 as a result. What am I doing wrong?

That's not how you need to calculate the partial sums. For your series,
S₁ = -1
S₂ = S₁ + 1 = 0
S₃ = S₂ - 1 = -1
...
and so on. As already mentioned, the partial sums oscillate between -1 and 0, so the series fails to converge in the usual sense. With alternating series, like this one, you can't group terms as you have done.

 

[jonjacson]

That's not how you need to calculate the partial sums. For your series,
S₁ = -1
S₂ = S₁ + 1 = 0
S₃ = S₂ - 1 = -1
...
and so on. As already mentioned, the partial sums oscillate between -1 and 0, so the series fails to converge in the usual sense. With alternating series, like this one, you can't group terms as you have done.

Could you justify why I cannot group terms as I did?

 

[Mark44]

Could you justify why I cannot group terms as I did?

Here's a quote from one of the Calculus texts I have, in Remark 4 on page 779 of "Calculus and Analytic Geometry, 2nd Ed.," by Abraham Schwartz:

It is shown in more advanced courses that the terms of an absolutely convergent series can be rearranged in any order and that the new series will converge again to the very same value. But if the terms of a conditionally convergent series are rearranged, the new series may or may not converge, and, if it dow converge, it may converge to a different value.

Some definitions:
A series $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges.
A series $\sum a_n$ is conditionally convergent if $\sum a_n$ converges but $\sum |a_n|$ diverges.

Your series, -1 + 1 - 1 + 1 + ... + ... is not absolutely convergent, because its partial sums increase without bound. Also, since the partial sums of your series oscillate (and therefore don't converge), the series isn't conditionally convergent, either. For these reasons, rearranging the terms in your series isn't valid.

 

[jonjacson]

Here's a quote from one of the Calculus texts I have, in Remark 4 on page 779 of "Calculus and Analytic Geometry, 2nd Ed.," by Abraham Schwartz:

Some definitions:
A series $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges.
A series $\sum a_n$ is conditionally convergent if $\sum a_n$ converges but $\sum |a_n|$ diverges.

Your series, -1 + 1 - 1 + 1 + ... + ... is not absolutely convergent, because its partial sums increase without bound. Also, since the partial sums of your series oscillate (and therefore don't converge), the series isn't conditionally convergent, either. For these reasons, rearranging the terms in your series isn't valid.

Ok, so this lead us to more advanced courses. I will search for those to find the answer.

 

[FactChecker]

As hilbert2 said

you can assign that sum ANY integer value by just rearranging terms.

There is no answer that can be considered authoritatively correct. It is fundamental in math to be able to rearrange a summation without changing the answer. But you can cluster any number of +1s or -1s together to get any integer values repeating as often as you want. So there is no well defined limit.

 

[jonjacson]

As hilbert2 saidThere is no answer that can be considered authoritatively correct. It is fundamental in math to be able to rearrange a summation without changing the answer. But you can cluster any number of +1s or -1s together to get any integer values repeating as often as you want. So there is no well defined limit.

Hi,

Could you show me how to rearrange the terms to get +2 as an answer?

Thanks

 

[WWGD]

Hi,

Could you show me how to rearrange the terms to get +2 as an answer?

Thanks

1+1 +(1-1)+(1-1)+...

 

[jonjacson]

1+1 +(1-1)+(1-1)+...

I am sorry but that does not correspond to the series I was talking about and I will explain why, let's take ANY quantity of numbers from this series, let's say 10:

-1, +1, -1, +1, -1, +1, -1, +1, -1, +1 As we see the proportion of -1 is the same as +1, and for every positive one there is a corresponding negative

if we now take 100 numbers the proportion will be the same, you will have 50 positive ones and 50 negative ones.

If we take any quantity of numbers that proportion is always constant.

Your mistake is you forget that in that series there must be -1 -1 somewhere else, that are the couple for your +1+1.

 

[FactChecker]

how to rearrange the terms to get +2 as an answer?

It's not a question of 2 being the "answer". There is no answer to that series. You can get 2 to show up over and over as terms are added.
You always have - 1 + 1 - 1 + 1 = + 1 + 1 - 1 - 1. So:
- 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...
= (- 1 + 1 - 1 + 1) + (- 1 + 1 - 1 + 1) + (- 1 + 1 - 1 + 1) + ...
Make the replacement of - 1 + 1 - 1 + 1 with + 1 + 1 - 1 - 1 an infinite number of times without changing the "answer" to get:
= (+ 1 + 1 - 1 - 1) + (+ 1 + 1 - 1 - 1) + (+ 1 + 1 - 1 - 1) + ...
= + 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ...
As you add term after term, you get the sequence of sums: 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, ...
So 2 shows up an infinite number of times. So does 1 and 0.
There is no answer to the sum of this series.

 

[WWGD]

But this is not so when you have an infinite collection of numbers. There is no clear cut proportion when you have an infinite collection; the best you can aim for is to have a bijection between the 1's and the -1's , which still holds if you pair numbers as I did.

Basically, you can swap , say the 1st and 4th terms in the series, and then even the proportion
will remain constant :

Original: -1+1 -1+1 -... Swap 1st and 4th to get:

1+1 -1 -1 + ( -1+1 -... )

And the proportion remains constant.

EDIT: I don't mean to imply that the series actually formally converges to 2, but that we can

rearrange the terms to have an informal sum equal to 2 ( or to 3, 4, etc. ).

 

[jonjacson]

It's not a question of 2 being the "answer". There is no answer to that series. You can get 2 to show up over and over as terms are added.
You always have - 1 + 1 - 1 + 1 = + 1 + 1 - 1 - 1. So:
- 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 + 1 ...
= (- 1 + 1 - 1 + 1) + (- 1 + 1 - 1 + 1) + (- 1 + 1 - 1 + 1) + ...
Make the replacement of - 1 + 1 - 1 + 1 with + 1 + 1 - 1 - 1 an infinite number of times without changing the "answer" to get:
= (+ 1 + 1 - 1 - 1) + (+ 1 + 1 - 1 - 1) + (+ 1 + 1 - 1 - 1) + ...
= + 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + 1 + 1 - 1 - 1 + ...
As you add term after term, you get the sequence of sums: 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, ...
So 2 shows up an infinite number of times. So does 1 and 0.
There is no answer to the sum of this series.

What you did basically is to show the sum of the series S1= -1+1-1+1-1+1-1... is equal to the series S2= -2 + 2 -2+2 -2+2-2+2 and your reasoning looks good to me, but that is consistent with what I said, that the sum of that series is equal to 0.
Since for every -2 you have a 2, and the sum of this series is S2=(-2+2) + (-2+2)... = 0 +0+0 = 0 since if we add an infinite amount of zeros the result is 0. What am I doing wrong?

But this is not so when you have an infinite collection of numbers. There is no clear cut proportion when you have an infinite collection; the best you can aim for is to have a bijection between the 1's and the -1's , which still holds if you pair numbers as I did.

Basically, you can swap , say the 1st and 4th terms in the series, and then even the proportion
will remain constant :

Original: -1+1 -1+1 -... Swap 1st and 4th to get:

1+1 -1 -1 + ( -1+1 -... )

And the proportion remains constant.

I don't understand why the proportion of -1 vs +1 does not hold when the set is infinite, Can you prove that?

 

[WWGD]

I don't understand why the proportion of -1 vs +1 does not hold when the set is infinite, Can you prove that?

EDIT: I think my statement was kind of confusing.
But the proportion actually does hold, just not at every single step of the way: you are just rearranging the order of the terms, so there
are as many -1's as there are 1's, only that these 1's, -1's are in a different spot on the sum.
The infinite number of terms allows you to add an infinite string of (1-1)'s to have a sum "equal" to 2.

 

[jonjacson]

Actually I am confused XD

I don't understand anything, Why (1-1)'s is "equal" to 2?

I said is equal to 0, not to 2.

 

[PeroK]

So the answer is that there is not a definite answer, in case we can associate a number we could pick the 1/2 number. Is that right?

I wonder what I am doing wrong with this reasoning:

1.- I think the answer is 0, since we can arrange the infinite series in pairs, for every -1 there will always be a 1, (-1 + 1) +(1-1) etc. As every one of those pairs sums 0, the total sum of the infinite seres should be 0. What is wrong? Why is it better 1/2 than 0?

Let S = -1 + 1 - 1 + 1 - 1 ... = 0

1 + S = 1 - 1 + 1 - 1 + 1 ... = -S = 0

So:

1 + 0 = 0

 

[jonjacson]

Let S = -1 + 1 - 1 + 1 - 1 ... = 0

1 + S = 1 - 1 + 1 - 1 + 1 ... = -S = 0

So:

1 + 0 = 0

I don't understand why it is equal to -S nor 0.

To me S= Infinite sum -1, +1, -1+1... if you add 1 to S you just have 1+ S = 1 because S is 0.

 

[FactChecker]

What you did basically is to show the sum of the series S1= -1+1-1+1-1+1-1... is equal to the series S2= -2 + 2 -2+2 -2+2-2+2 and your reasoning looks good to me, but that is consistent with what I said, that the sum of that series is equal to 0.
Since for every -2 you have a 2, and the sum of this series is S2=(-2+2) + (-2+2)... = 0 +0+0 = 0 since if we add an infinite amount of zeros the result is 0.

Ha! I see what you are saying. But the way the total of an infinite sum is defined, you must look at each partial sum after adding one term at a time and those partial sums must eventually converge to a single total number. The series of partial sums of my final summation are 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0 ... So the partial sums are not converging to a single number and never will.

There are many ways that your series can be manipulated to give wild swings in the partial sums. That is because it has an infinite total of positive numbers and an infinite total of negative numbers that you can intersperse in the summation to make the partial sums keep changing any way you want.

That is not always true. If the sum of all the positive numbers is finite and the sum of all the negative numbers is finite, there will be a single number that the partial sums converge to, no matter how you rearrange them. That is because you have a limited amount that you can manipulate the series to swing the partial sums around. When your plus total or minus total starts to run out, the partial sums eventually converge to the answer.

 

[PeroK]

I don't understand why it is equal to -S nor 0.

To me S= Infinite sum -1, +1, -1+1... if you add 1 to S you just have 1+ S = 1 because S is 0.

If S = -1 + 1 - 1 + 1 ... = 0

So T = 1 -1 + 1 -1 ... = 0 also?

Why isn't?: 1 + (-1 + 1 - 1 + 1 ...) = 1 - 1 + 1 - 1 ...

 

[jonjacson]

Ha! I see what you are saying. But the way the total of an infinite sum is defined, you must look at each partial sum after adding one term at a time and those partial sums must eventually converge to a single total number. The series of partial sums of my final summation are 1, 2, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0 ... So the partial sums are not converging to a single number and never will.

There are many ways that your series can be manipulated to give wild swings in the partial sums. That is because it has an infinite total of positive numbers and an infinite total of negative numbers that you can intersperse in the summation to make the partial sums keep changing any way you want.

That is not always true. If the sum of all the positive numbers is finite and the sum of all the negative numbers is finite, there will be a single number that the partial sums converge to, no matter how you rearrange them. That is because you have a limited amount that you can manipulate the series to swing the partial sums around. When your plus total or minus total starts to run out, the partial sums eventually converge to the answer.

Why? Why must I look one by one? Why can't I just put two terms together?

I mean, let's forget series, Isn't this +1-1 equal to (+1-1)?

 

[jonjacson]

If S = -1 + 1 - 1 + 1 ... = 0

So T = 1 -1 + 1 -1 ... = 0 also?

Why isn't?: 1 + (-1 + 1 - 1 + 1 ...) = 1 - 1 + 1 - 1 ...

If you mean T= 1+S= 1 ; I agree since if you add an extra one to a series that contains the same number of -1 as +1 , you are really adding something extra and that is different to those "rearrangements" where they showed more +1 +1 than -1 just because they wanted to forget that for every +1 there is a -1 in the series.

 

[PeroK]

If you mean T= 1+S= 1 ; I agree since if you add an extra one to a series that contains the same number of -1 as +1 , you are really adding something extra and that is different to those "rearrangements" where they showed more +1 +1 than -1 just because they wanted to forget that for every +1 there is a -1 in the series.

So, S = -1 + 1 - 1 + 1 ... = 0

And T = 1 - 1 + 1 - 1 ... = 1

So, why isnt't T = -S?

 

[jonjacson]

I am sorry but T and S are completely different things:

T= +1+1-1+1-1+1-1+1-1+1-1
S= +1-1+1-1+1-1+1-1+1-1
T=S+1 and is different to -S

you just rearranged the series in a way that was misleading since it didn't show that you actually have added an extra +1 to S.

 

[PeroK]

I am sorry but T and S are completely different things:

T= +1+1-1+1-1+1-1+1-1+1-1
S= +1-1+1-1+1-1+1-1+1-1
T=S+1 and is different to -S

you just rearranged the series in a way that was misleading since it didn't show that you actually have added an extra +1 to S.

Okay, let's start again. What series are you saying is equal to 0?

 

[jonjacson]

S=0=...+1-1+1-1...

 

[PeroK]

S=0=...+1-1+1-1...

Okay, what about?

T = -1 + 1 - 1 + 1 ...

 

[PeroK]

I've got to go, so let me make my point:

a) T = 0 by the same argument you used for S. Just group the terms.

But:

b) S = 1 + T. So, if T = 0, then S must be 1.

So, something must have gone wrong.

 

[jonjacson]

I've got to go, so let me make my point:

a) T = 0 by the same argument you used for S. Just group the terms.

But:

b) S = 1 + T. So, if T = 0, then S must be 1.

So, something must have gone wrong.

We can continue talking another day, no problem.
a) T=1 because you forget that S contains a -1 for every +1, and that proportion holds no matter the size of the set. If you add +1 to create T there is no -1 as a counterparty, that is why T sums +1 and S sums 0, because you added an extra +1 to S but you did NOT add the corresponding -1 that would maintain the same proportion of +1 and -1.

 

[PeroK]

We can continue talking another day, no problem.
a) T=1 because you forget that S contains a -1 for every +1, and that proportion holds no matter the size of the set. If you add +1 to create T there is no -1 as a counterparty, that is why T sums +1 and S sums 0, BECAUSE YOU ADDED AN EXTRA 1 that did not have his corresponding -1 in the series.

How do you know that S does not have an extra 1?

T = -1 + 1 -1 +1 ... = (-1 + 1) + (-1 + 1) ... = 0 + 0 + 0 ... = 0

So, T has just as much right to be 0 as S has. Why prefer S? Just because it starts with a positive number? I might prefer T = 0 and S = 1.

Also, why isn't T = -S?

Surely if you mutiply S by -1 you get T? So, how can S = 0 and T = -1?

 

[jonjacson]

How do you know that S does not have an extra 1?

T = -1 + 1 -1 +1 ... = (-1 + 1) + (-1 + 1) ... = 0 + 0 + 0 ... = 0

So, T has just as much right to be 0 as S has. Why prefer S? Just because it starts with a positive number? I might prefer T = 0 and S = 1.

Also, why isn't T = -S?

Surely if you mutiply S by -1 you get T? So, how can S = 0 and T = -1?

The first is a good question. Because that would imply a kind of assimetry, if there is an extra 1 it means the amount of -1 is less than the amount of +1. But why isn't it an extra -1 instead of a +1?

It doesn't make sense, the only consistent answer to that question is that the amount of +1 in the series is equal to the amount of -1 in it.

In other words, for every +1 in the series we can find its corresponding -1, and that is exactly what we see if we look at the terms in that series, you can take 10, 100, or a million terms and you will always see there is a -1 for every +1.

Every time you represent T you forget to show you added an extra +1 to S, let me show that extra +1 and all your confusions will dissapear:

T= ...-1+1-1+1+1-1+1-1+1
S=...-1+1-1+1-1+1-1+1-1

As you see there is an "anomaly" in T and is precisely the extra +1 you added to S. With that representation all the answers to your questions are simple:

T=S+1=1

I see you insist in multiplying -1 by S to get -S.

Let's do the same for T, we get:

-T=...+1-1+1-1-1+1-1+1-1... =-1 =-s-1 or in other words, if you add an extra -1 to -S you get -T which sums -1 instead of 0

 

[FactChecker]

Why? Why must I look one by one? Why can't I just put two terms together?

I mean, let's forget series, Isn't this +1-1 equal to (+1-1)?

Yes, they are equal.
Suppose we allow -1+1-1+1... to be a valid infinite summation with a single answer.
On one hand, we have -1 +(+1-1) + (+1-1) +(+1-1) + ... = -1 +0 +0+0...= -1
On the other hand we have (-1+1) + (-1+1) + (-1+1) + ... = 0+0+0... = 0
Which one is correct? This violates the associative property of addition, which is fundamental to all mathematics.

In math, a summation has a valid total only if changing the order or clustering of terms does not change the number that the partial sums eventually converge to. The associative and commutative properties of addition are not violated by the math definition of an infinite sum.

We either have to give up on -1+1-1+1... having a well-defined total, or we have to give up the associative and commutative properties of addition. The choice is clear. You will see eventually that "summations" like -1+1-1+1... are not very interesting and we haven't given up much.

 

[jonjacson]

Yes, they are equal.
Suppose we allow -1+1-1+1... to be a valid infinite summation with a single answer.
On one hand, we have -1 +(+1-1) + (+1-1) +(+1-1) = -1 +0 +0+0...= -1
On the other hand we have (-1+1) + (-1+1) + (-1+1) + ... = 0+0+0... = 0
Which one is correct? This violates the associative property of addition, which is fundamental to all mathematics.

In math, a summation has a valid total only if changing the order or clustering of terms does not change the number that the partial sums eventually converge to. The associative and commutative properties of addition are not violated by the math definition of an infinite sum.

We either have to give up on -1+1-1+1... having a well-defined total, or we have to give up the associative and commutative properties of addition. The choice is clear. You will see eventually that "summations" like -1+1-1+1... are not very interesting and we haven't given up much.

The second one is correct and it is equal to 0.

Since in the first case you are telling us there are extra +1 that do not have their corresponding -1. And my question is, Why only one? Why don't you add magically 25 extra +1? To me you are doing this mistake:

+1+1+1+1+1+1+1 +(+1-1) + (+1-1) + (+1 - 1) = 7 +0+0= 7 ; what I have been telling since the first post is that for those seven +1 there must be in the series their corresponding seven -1, what confuses you is you don't write them, so it looks like they don't exist and the result is absurd.

What I think is accepting the sum is = 0 does not generate any inconsistency nor absurd result and I would like you to demonstrate where I am wrong since I am not even rearranging terms. I have a question for you, assuming the reesult is 0, Does it violate any mathematical rule like associative?

edit:

I mean accepting the series =0 as the only valid result and all the other ones are incorrect.

 

[FactChecker]

Since in the first case you are telling us there are extra +1 that do not have their corresponding -1.

No. They are the exact same numbers in the exact same order. The associative law has just been applied differently in the two cases. By the associative property, that is not allowed to give different answers.

 

[ImperialThinker]

The series is not convergent though it is bounded. So the series sum cannot be determined.
However, in case of divergent series we don't usually determine the entire sum, the ramanujan sum of the series is 1/2

 

[PeroK]

Every time you represent T you forget to show you added an extra +1 to S, let me show that extra +1 and all your confusions will dissapear:

T= ...-1+1-1+1+1-1+1-1+1
S=...-1+1-1+1-1+1-1+1-1

One last try, the I give up!

Suppose you had never posted this thread and I defined T = -1 + 1 -1 +1 ... (without any reference to S). Why can't I conclude that T = 0?

Then you come along with S = 0. Well, your series is just T with a 1 at the front. I would then say S = 1 + T. So, I would say T = 0 and S = 1. And, you would say S = 0 and T = -1. But, we would only be saying this because of the series we had calculated first.

(In fact, if you look at your original post, you DID specify T. The original series you posted started with a -1, not with a 1. So, which is it? Which one is "really" the series that adds to 0 and which is the sequence that adds to 0 with an extra term at the front?).

Also, in any case, why isn't:

S = 1 - 1 + 1 - 1 ...
T = -1 + 1 - 1 + 1 ...
S+T = 0 + 0 ... = 0

Finally, what about

S' = a - a + a - a ... = 0

You're saying that S' = 0, but only when a is positive? But surely you can group a - a = 0 for any a, regardless of whether a is +/-?

Surely S' = 0 for any a?

 

[zoki85]

Who says that every infinite series must have sum?

 

[Mark44]

In math, a summation has a valid total only if changing the order or clustering of terms does not change the number that the partial sums eventually converge to. The associative and commutative properties of addition are not violated by the math definition of an infinite sum.

What I think is accepting the sum is = 0 does not generate any inconsistency nor absurd result and I would like you to demonstrate where I am wrong since I am not even rearranging terms.

But you are grouping the terms in a certain way. In an absolutely convergent series, reordering or grouping don't change the sum. The series in question here is not absolutely convergent, so reordering and grouping aren't valid.

I have a question for you, assuming the reesult is 0, Does it violate any mathematical rule like associative?

You can't just assume that the series in question is convergent.

The question has been asked and answered, so I am closing this thread.