Thread: Why is the sum of 1-2+3-4+... is 1/4?

1. Why is the sum of 1-2+3-4+... is 1/4?

Can someone explain to me the paradox 1-2+3-4+... =1/4? I really don't get why and how. Thanks so much.

2. Originally Posted by chengbin
Can someone explain to me the paradox 1-2+3-4+... =1/4? I really don't get why and how. Thanks so much.
The binomial series $\displaystyle 1-2x+3x^2-4x^3+\ldots$ converges to $\displaystyle (1+x)^{-2}$ when $\displaystyle |x|<1$. If you ignore that restriction on x and pretend that the series also converges to the same function when x=1 then you come up with the formula $\displaystyle 1-2+3-4+\ldots = 1/4$, which of course is not true in any conventional sense.

3. Call $\displaystyle S:= 1-1+1-\cdots$. Cancelling off the very first term by subtracting 1 gives $\displaystyle -1+1-1+\cdots= - (1-1+1-\cdots)$ so $\displaystyle S = -(S - 1)$ or $\displaystyle S=\frac12$.

Call $\displaystyle T := 1-2+3-4+\cdots$. Add the series $\displaystyle 1-2+3-4+\cdots$ and $\displaystyle 1-1+1-1+\cdots$ together term by term to get $\displaystyle T+S = (1+1) - (2+1) + (3+1) - \cdots = 2 - 3 + 4 -\cdots = -(T-1)$. Since $\displaystyle S=\frac12$, $\displaystyle T+\frac12 = -(T-1)$ or $\displaystyle T=\frac14$.

The remarkable thing, of course, is that you get the same answer if you do it "algebraically" like this as if you do it like Opalg does it (analytically, generating funtions).

4. Originally Posted by chengbin
Can someone explain to me the paradox 1-2+3-4+... =1/4? I really don't get why and how. Thanks so much.
$\displaystyle 1-2+3-4+\dots = (1 - 2) + (3 - 4) + (5 - 6) + (7 - 8) + \dots$

$\displaystyle = (-1) + (-1) + (-1) + (-1) + \dots$

$\displaystyle = \lim_{n \to \infty}\frac{n}{2}(-1)$.

This is clearly not $\displaystyle \frac{1}{4}$...

5. There is no way this is 1/4. The reason? Because, integers are closed under addition.

6. Any convergent series whose terms are integers has only finitely many non-zero terms. But closure only applies to finite sums. For instance, the rationals are closed under addition but the sum $\displaystyle 1+\frac1{2!}+\frac1{3!}+\frac1{4!}+\cdots$ is irrational.

7. Looks like I need to read up on my Real Analysis again, because this is not making sense to me.

8. Lvleph is right in a sense. Since you only consider the addition and the substraction of integer terms, you cannot possibly end up with a non-integer result. So the original statement cannot be right !

1 ? 2 + 3 ? 4 + · · · - Wikipedia, the free encyclopedia

I don't know enough math to understand the "proofs" in that article. Can anyone explain the number manipulation method? It seems like the proof that requires the least amount of math knowledge to understand.

10. Originally Posted by Bacterius
Lvleph is right in a sense. Since you only consider the addition and the substraction of integer terms, you cannot possibly end up with a non-integer result. So the original statement cannot be right !
Things get weird when we begin talking infinity though. But then again, the abstract algebra sense of the Integers does include an infinite collection. Even the linear algebra sense gives you closure since the Integers are a vector space.

EDIT: I can see how this would be 1/4 in the formal power sense (which is what maddas was saying), but this sub forum is not that advanced.

11. Nothing else to do, so I'm trying:
1 - 2 - 3 + 4 + 5 - 6 - 7 + 8 + 9 -10 - 11 + 12 + 13 ....

Well, this gives me running totals of 1, -4, 5, -8, 9, -12, 13 ...

Now how in heck can I paradoxize this

12. Originally Posted by lvleph
There is no way this is 1/4. The reason? Because, integers are closed under addition.

This is a very bad reason: also rationals are closed under addition yet $\displaystyle \sum^\infty_{k=0}\frac{1}{n!}=e\notin\mathbb{Q}$ ...

Tonio

13. Originally Posted by Bacterius
Lvleph is right in a sense. Since you only consider the addition and the substraction of integer terms, you cannot possibly end up with a non-integer result. So the original statement cannot be right !

Nop, this is not a good reason: the conterexample $\displaystyle \sum^\infty_{k=0}\frac{1}{n!}=e\notin\mathbb{Q}$ shows that a sum of only rational numbers can have a non-rational sum...of course, as already noted, the gist here is that the sum is infinite and thus closedness and other related things do not apply here.

Tonio

14. Originally Posted by lvleph
Things get weird when we begin talking infinity though. But then again, the abstract algebra sense of the Integers does include an infinite collection. Even the linear algebra sense gives you closure since the Integers are a vector space.

They are? Under what field, say?

Tonio

EDIT: I can see how this would be 1/4 in the formal power sense (which is what maddas was saying), but this sub forum is not that advanced.
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15. The value of the series depends on the method used to sum it. Everyone knows that the series diverges in the traditional sense (aka. the partial sums are eventually graeater in magnitude than any number). The series is Abel summable to 1/4. The wikipedia article the OP linked discusses other summability methods under which it converges. In some sense, any "reasonable" summation method which sums this series must give 1/4. Hardy has a book on divergent series for anyone interested in these sums and their applications.

(Also, Wilmer, you have wrong series; the signs alternate every term, not every two terms. The partial sums are actually 0,1,-1,2,-2,3,-3,...)

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