1 + 2 + 4 + 8 + 16 . . . = -1

[simonm]

Let x = 1 + 2 + 4 + 8 +16 . . .

Then x-1 = 2 + 4 + 8 + 16 . . .

and (x-1)/2 = 1 + 2 + 4 + 8 . . .

so (x-1)/2 = x

so x - 1 = 2x

so -x = 1

and x = -1

so

1 + 2 + 4 + 8 + 16 . . . = -1

[Guv]

Simonm: Very cute.

It shows that one must be careful when dealing with divergent series. I have seen several threads reaching erroneous conclusions due to the use of unallowable operations on divergent infinite series.

[alkatran]

and (x-1)/2 = 1 + 2 + 4 + 8 . . .

so (x-1)/2 = x

hold on, so..

(1 + 2 + 4 + 8 +16 . . . -1)/2 = 1 + 2 + 4 + 8 +16 . . .

I don't tihnk that makes sense, (Grade 11 here)

I prefer this:

(5-5)4 = (5-5)1

Cancel 5-5 on both sides

4=1















I divided by 0, me so bad (got that one from bugz).

[kedaman]

this reminds me of signed integers

[starwiz]

(2 + 4 + 8 + 16...) / 2 != 1 + 2 + 4 + 8 ...that's apparent: fine...

But why can't we apply the idea of (2+4) / 2 = 2/2 + 4/2 ?

Why are infinite series(es?) different?

[Guv]

Starwiz: As defined in the initial post, x is an unbounded or infinite number. It is equal to the sum of an unbounded series.

If infinite or transfinite numbers obeyed all the rules of arithmetic applied to ordinary numbers, the transfinite numbers would not be different from ordinary numbers. Obviously they are different.

Either you have to accept the notion that x is not an allowable number or accept that x/2 = x or accept some some other special rules for transfinite numbers. The alternative is being forced to believe in proofs for obviously silly results.

Do you really believe that 1 + 2 + 4 + 8 + 16 . . . = -1?

[starwiz]

Do you really believe that 1 + 2 + 4 + 8 + 16 . . . = -1?

Certainly not. But the proof that "this can't be true because it's obviously not true" doesn't seem too much like a proof, you know?

I was just asking: Infinite series are different than normal series--fine, that's obvious enough. But how do you prove it, other than "if they were, then this would be true, and it's not"? Or is that the proof?

[twanvl]

The way I see it:

x = 1 + 2 + 4 + 8 +16 . . . = Infinity
x-1 = Infinity
(x-1)/2 = Infinity
(x-1)/2=x == Infinity=Infinity

Infinity is not a real number, therefore you can't treat it like that.

[Guv]

Starwiz: There is a method of proof called Reductio ad absurdum, which is a fancy way of saying if the conclusion is silly, there is something wrong with an assumption used along the way.

One form of such proof is to start by assuming that a desired conclusion is false. Then you come up with some paradoxical conclusion (at least that is what to hope to do). When you get the silly conclusion, you say: therefore my assumption was incorrect, and the desired conclusion must be true.

I am not use exactly what is incorrect with the proof in this thread. It might be viewed as improper to equate x to the sum of an infinite series. It might be invalid to do ordinary arithmetic on x, which is a transfinite number. For such numbers, (x-1) = x, as well as 2*x = x.

Infinite numbers and infinite sets are strange animals. For instance, you can show that the set of all even integers and the set of all integers have the same number of members. Note that you cannot actually count how many members each set has. To determine which set has more members, you try to match up members from each set.

If you can make a one-to-one match up, they have the same number of members. If you can show that any attempt to match them up results in all the members of one set being used up while the other has some members unmatched, then the set with none left over is smaller. Try this with the set of all integers and the set of all even integers. The can be matched up as follows.

1 2
2 4
3 6
4 8
. . .
n 2n
(n+1) (2n+2)
. . .
The members of the two sets can be matched up with no members from either set unmatched. Therefore the two sets have the same number of members.

No matter how strange the above seems, it is certainly better than believing that

-1 = 1 + 2 + 4 + 8 + 16 . . .

[Spooner]

Originally posted by kedaman
this reminds me of signed integers

That was my first thought too. Because if you add them in binary, to a word length of your choice, you get 11111111111... which will indeed be -1:

00000001 : 1
00000010 : 2
00000100 : 4
00001000 : 8
00010000 : 16
00100000 : etc etc
01000000
10000000
________
11111111 = -1 in signed binary

[sql_lall]

it looks nice

however, i agree with twanvl.
x-1=2x

x = infinity, so you can't just 'take' x from both sides

cos infinity-1=infinity=2*infinity, so yeah, it is true, but you can't just cancel.

Yet, i will still have to try this out with my teachers and friends

[Guv]

Spooner: What moves the binary point an infinite number of places to the left?


.1111111111111111 . . . . is minus one

1111111111111111 . . . . is a biiiiig number

[kedaman]

 01111111111111111...1 (n 1's)
-1000000000000000...0 (n 0's)
_____________________
-1

the deal is that the infite+1'th digit was left out in simon's proof

[TereNCE19]

i not agree what the simonm show it.....

my idea is/.....

Let x = 1 + 2 + 4 + 8 +16 . . . n (n represent any integer numbe)

Then x-1 = 2 + 4 + 8 + 16 . . . n

and (x-1)/2 = 1 + 2 + 4 + 8 . . n/2

so how come (x-1)/2 = x ???

so (x-1)/2 not equal to x


it is not logically for 1+2+4+8+16...=-1

[starwiz]

Let x = 1 + 2 + 4 + 8 +16 . . . n (n represent any integer numbe)

This is changing the proof.

The proof shows an infinite series...there is no end to the numbers added together to x. You are changing it to a finite series; doing so could certainly change many aspects of the proof, including whether it's obviously false or not.

[TereNCE19]

x=1+2+4+8+16......n

i had make some mistake

n is not a number
coz x=1+2+4+8+16.....untill infinite

Infinite is not a number....can not divide or do something....this is
a defination

so

(x-1)/2 not equal to x

[marnitzg]

Heh heh. Reminds me of the good old 0.9(rec) = 1 thread

[DiGiTaIErRoR]

x = 1+2+4+8+16... = infinity

infinity - 1 = infinity

---
so x - 1 = 2x

so -x = 1
---

ok...

let x = infinity

x-y=2x
x = infinity = y

While other numbers can be plugged in for y, the only true y is infinity.