Why is this wrong? (1 = 2 proof)

[Jack_Ian]

[wordcross]

When you first add the pieces in parentheses, you have an infinite number of sets. When you remove them, then you are ignoring the fact that they are sets. It should really look like this:

[Jack_Ian]

In infinite set ending in a -1?

I know the proof is wrong. My problem is that if I take any two consecutive lines, the step taken seems mathematically correct. It seems clear given the example above that the logic is incorrect, but how would I recognise such an error in a more complicated algebraic proof?

[mith]

http://mathworld.wolfram.com/ConditionalConvergence.html

[wordcross]

Alright fine, how about this way, then:

[Jack_Ian]

[Jack_Ian]

[wordcross]

Nope, that's fine. You still have a -1 for every +1 after the first.
_________________
Has anyone really been far even as decided to use even go want to do look more like?

[Nsof]

[MatthewV]

This is why people usually will show the first few and the last few terms.

[extro...*]

[extro...*]

Here: http://en.wikipedia.org/wiki/Grandi's_series

[extro...*]

[Jack_Ian]

Thank you extro…
That was exactly what I was looking for.

[Jack_Ian]

I showed my 8 year old daughter the above proof and asked her where she thought the mistake was made.
She told me that she wasn't sure about the third step, i.e. "1 = 1 + 1 + -1 + 1 + -1 + 1 + -1 + … ", and wanted to ask me a question first.
Sure I said, ready to display my brilliance and become her mathematical hero.
"Is infinity odd or even?"
I was stumped. I said it was neither, but I could tell she wasn't impressed with that answer.

[Chuck]

Let x = 1-1+1-1+1-1+1-1...

subtract 1 from each side

x-1 = -1+1-1+1-1+1-1+1...

add the above two equations together

2x-1 = 0+0+0+0+0+0+0... = 0

add 1 to each side

2x = 1

divide both sides by 2

x = ½

[ralphmerridew]

The short answer is that you have to justify that the sum of an infinite series is well-defined before you do anything related to that sum.

Doing otherwise is akin to to "Let P be the point where lines L1 and L2 intersect. For reason X, P must be equidistant from points A and B ..." without first proving that lines L1 and L2 do intersect.

[davidchatman]

Finally, here is the Nobel prize nomination for mathematics

[cb*]

[yevalent]

Here is the correct sequence ... 1=2 never possible .......

1 = 1 + 0 + 0 + 0 + …
1 = 1 + (1 + -1) + (1 + -1) + …
1 = 1 + 1 + -1 + 1 + -1 + 1 + …+ -1
1 = 1 + 1 + (-1 + 1) + (-1 + 1) + … + -1
1 = 1 + 1 + 0 + 0 + 0 + … + -1
1 = 1 (Hence Proved) ....

[extropalopakettle]

1 = 1 + (1 + -1) + (1 + -1) + …
1 = 1 + 1 + -1 + 1 + -1 + 1 + …+ -1

How can you just stick a "+ -1" at the end?