Infinite sums (mathematics help)

[MisterM2402]

I'm reading one of the appendices in my maths textbook for university and it is explaining Sigma Notation. It was recommended to do this by my lecturer since I didn't take the more advanced maths course in the last year of high school (we call those courses "Advanced Highers" here in Scotland, that follow on from "Highers").

One part of the book says that infinite sums (summing an infinite sequence of terms) don't always have a value and that great care must be taken when rearranging or manipulating their terms. They give the example of

S = 1 + 2 + 4 + 8+...
= 1 + 2(1 + 2 + 4 +...)
= 1 + 2S
--from which is follows that S = -1. This is clearly nonsense, since S is a sum of non-negative terms! (Where is the error?)

I have no idea where the error is and it's frustrating that they don't give you the answer even at the very back of the book XD. I've been puzzling this for quite a while now and I've had no luck.

I also have no idea how to show what the sigma notation that they started off with looked like. Is it even possible on this forum? Not sure how it would be written in mathematical English either :/

Thanks a lot.

[I'm betting nobody will believe me when I say that this isn't homework (it really isn't, it's just extra reading), so could you at least give a hint as to the answer, please?]

 

[Zantos]

This is essentially an exercise in "Think about it". The error is intrinsic to the fact that in an infinite sum you cannot under any circumstances not even once by mistake if you come in drunk and forget which bed is yours or worse piss in the fridge assign parentheses like that. Essentially the way to think about it is that if it's an infinite sum there is no final number to put the closing parenthesis after, so the 2*(1+2+....) doesn't exist. It's kind of wishy washy, but the formal proof is a little more, disconcerting. Also I can't find it.

You can do something almost similar with convergent series by defining them as finite series to within a small error, but that's a whole different kettle of fish.

 

[DasDestroyer]

Since S is infinity, normal number logic doesn't apply.
If you limit it, thereby restoring the sanity of the whole situation, you end up with more or less
S = 1 + 2 + 4
S = 1 + 2(1 + 2)
Which obviously makes sense
Edit:
From what I recall this is pretty much identical to the division by zero mistakes, since, AFAIR, x/0 is sometimes assumed to be infinity.

0*1=0
0*2=0
therefore
0*1=0*2
divide both sides by zero
1=2
Pretty obvious where the error is.

Similarly, but related to infinite sums:
0=0+0+0+0+0+0+...
0=(1-1)+(1-1)+(1-1)...
0=1-1+1-1+1-1+1...
0=1+(-1+1)+(-1+1)+(-1+1)...
0=1

 

[MisterM2402]

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Thank you guys so much; your explanations make SO much sense haha! (Didn't mean that to sound sarcastic). It seems such a simple answer, but I just couldn't see it :/

Been fretting over this for a few days now, good to put my mind at ease

 

[Zantos]

This is essentially an exercise in "Think about it". The error is intrinsic to the fact that in an infinite sum you cannot under any circumstances not even once by mistake if you come in drunk and forget which bed is yours or worse piss in the fridge assign parentheses like that. Essentially the way to think about it is that if it's an infinite sum there is no final number to put the closing parenthesis after, so the 2*(1+2+....) doesn't exist. It's kind of wishy washy, but the formal proof is a little more, disconcerting. Also I can't find it.

You can do something almost similar with convergent series by defining them as finite series to within a small error, but that's a whole different kettle of fish.

Eh, the parenthesis are technically correct. And it works if the sum converges.

Example:

S = 1 + 1/2 + 1/4 + 1/8 +...
S = 1 + 1/2(1 + 1/2 + 1/4 + 1/8 + ...)
S = 1 + 1/2 S
S - 1/2 S = 1
1/2 S = 1
S = 2

You can use it to derive the formula for a geometric series whose constant ratio is less than 1.

S = a + ar + ar^2 + ar^3 + ...
S = a + r(a + ar + ar^2 + ...)
S = a + rS
S - rS = a
S = a/(1-r)

Though I usually see it done by declaring rS = ar + ar^2 + ... and going from there, but rS has an implied use of parenthesis anyway. So, no, the parenthesis aren't a real problem.

That only works for convergent series though, of which that example is definitely not. My options were either try and explain it in a way that wasn't formally correct but easier to visualise, or break out the pure maths books and get my epsilons in there. I decided against epsilons.