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

**mohammadfawaz** · April 25th 2010, 09:34 AM

Clearly this diverges by nth term test. But what i'm not understanding is that the proofs provided on wikipedia make sense also! they even provide a proof to the following: $1-1+1-1+1-1+... = \frac{1}{2}$ !!! HOW IN THE WORLD CAN THIS BE EXPLAINED?

**Wilmer** · April 25th 2010, 10:53 AM

Originally Posted by maddas

(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,...)

NO I don't: I'm crearing my own!
Will be known as The Wilmer Paradox.
I'll become famous, and will be buried next to Jethro Clampett...

**Bacterius** · April 25th 2010, 11:04 AM

Originally Posted by chengbin

Read this

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.

What ? Now I must tell you that I am not convinced at all. Your expression is equivalent to :

$\sum_{n = 1}^{\infty} -n(-1)^n$

And this sum diverges. So how can it be equal to $\frac{1}{4}$ ? Unless there is some secret mathematical trick that I am not aware of, I don't understand what's up with this statement which is by any means wrong.

Any clues ?

**maddas** · April 25th 2010, 11:06 AM

@Wilmer

lol

Well, without any semblance of rigour, write $S:=-1-1+1+1-1-1+\cdots$ . It satisfies $-(S+1+1) = S$ so $S=1$ . Let $T:=-2-3+4+5-6-7+\cdots$ . Adding these term by term gives $T+S = -3-4+5+6-7-8+\cdots$ . Adding S again gives $T+2S = -4-5+6+7-8-9+\cdots = -2-3-T$ . Therefore $T=-7/2$ , if I have made no error. Therefore the Wilmer paradox is $1-2-3+4+5-\cdots = 1-\frac 72 = -\frac 52$ :]

**maddas** · April 25th 2010, 11:08 AM

Originally Posted by Bacterius

Any clues ?

Yes, read the wikipedia article.

**Bacterius** · April 25th 2010, 11:12 AM

Originally Posted by mohammadfawaz

Clearly this diverges by nth term test. But what i'm not understanding is that the proofs provided on wikipedia make sense also! they even provide a proof to the following: $1-1+1-1+1-1+... = \frac{1}{2}$ !!! HOW IN THE WORLD CAN THIS BE EXPLAINED?

Okay let's do this one quickly.

$1 - 1 + 1 - 1 + 1 - \cdots = (1 - 1) + (1 - 1) + 1 - \cdots = 0 + 0 + 1 - \cdots \color{red}{\neq \frac{1}{2}}$

Well, that's what I think. I don't care what all those integrals on Wikipedia tell me, for me this sequence is definitely equal to $1$ for odd $n$ and $0$ for even $n$ , regardless of the value of n. Even if n goes towards infinity.

No one can provide a proof for something that makes no sense. I don't have such a high level of abstraction, sorry. I'm just going to leave this one unsolved because this is getting really confusing. It's a paradox after all. And I don't seem to be prepared to embrace the fact that standard mathematics are not enough to answer this.

**maddas** · April 25th 2010, 11:17 AM

We are not talking about the partial sums. See Divergent series - Wikipedia, the free encyclopedia

**Bacterius** · April 25th 2010, 11:24 AM

Originally Posted by maddas

We are not talking about the partial sums. See Divergent series - Wikipedia, the free encyclopedia

Okay maddas, now I'm going to ask you something.
The OP's sum is :
- divergent under standard tests.
- convergent using other methods.

How can a sum possibly be divergent and convergent ? There must be one of the methods used that fails. Unless I fail. But I'm going to give up because I feel I suck at this kind of mathematics. I'm not even able to understand this stuff even after reading twice the Wikipedia article

**lvleph** · April 25th 2010, 11:27 AM

It depends on what space you are talking about. Another good example is
$\lim_{n \to \infty} \sin nx$
This has has no limit in the normal sense, but in the sense of distributions the limit is zero.

**Bacterius** · April 25th 2010, 11:32 AM

Originally Posted by lvleph

It depends on what space you are talking about. Another good example is
$\lim_{n \to \infty} \sin nx$
This has has no limit in the normal sense, but in the sense of distributions the limit is zero.

What is $x$ ? Is it a constant ?

**lvleph** · April 25th 2010, 11:41 AM

$x\in \mathbb{R} \backslash\{0\}$

**mohammadfawaz** · April 25th 2010, 12:05 PM

So the definition of the sum of a series as the limit of the partial sum is not actually correct? Or not always correct? I that sense, what we know about summation of series and all the tests we use are based on that definition. For the given series to converge, there must be another definition to use!

**maddas** · April 25th 2010, 12:11 PM

A definition cannot be correct or incorrect. It can only be useful, or natural, or motivated, or ...

This thread really went off the deep end at some point.

**Wilmer** · April 25th 2010, 01:05 PM

Originally Posted by maddas

This thread really went off the deep end at some point.

...at the beginning

**lvleph** · April 25th 2010, 01:06 PM

It's my fault.

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

LinkBack

Thread Tools

Display

Search Tags