What is wrong with this?

[skip0110]

Well, what is?

0 = 0 + 0 + 0 + 0 + 0 + 0 + 0 + ...
  = (1-1) + (1-1) + (1-1) + (1-1) + ...
  = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
  = 1

Clearly sometime this type of grouping is valid (telescoping sums, like sum [1/(n+1) - 1/n] from 1 to infinity) and sometimes it is not.
I have my own ideas, but I'd like to hear a few other opinions on the matter first.

 

[Omnis]

= 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) + ...
= 1

Nothing wrong with the problem. One equals one.

 

[skip0110]

But I started with zero!

Look carefully, between lines 2 and 3 I simply regoruped the parentheses.

 

[Omnis]

Well, duh. By doing that, you mess up the whole order of operations. Plus, there is still the 1 hanging out there alone. For a true regrouping, wouldn't you need a 1 hanging out somewhere on line 2?

 

[Sage]

Parentheses make all the difference in the world when subtracting.

[font=Courier, Courier New, Monospace, monospace]9 – (7 + 10) = -8
(9 – 7) + 10 = 12[/font]

So it’s not just applicable to infinite series.

 

[skip0110]

Well, duh. By doing that, you mess up the whole order of operations. Plus, there is still the 1 hanging out there alone. For a true regrouping, wouldn't you need a 1 hanging out somewhere on line 2?

Add and subtract are on "equal footing" so to say, so I can definitely move parentheses around as I like.

And since I have an infinte number of ones, adding one or taking one out from the group shouldn't matter! Thats what they do to solve telescoping sums.

Parentheses make all the difference in the world when subtracting.

[font=Courier, Courier New, Monospace, monospace]9 – (7 + 10) = -8
(9 – 7) + 10 = 12[/font]

So it’s not just applicable to infinite series.

I had (1-1) + (1-1) + (1-1) + (1-1) + (1-1) + (1-1) + .... which is the same with or without parentheses.

 

[Omnis]

Whoops. I missed where he acknowledged that it was indeed invalid. But, still, you can't change groupings without maintaining equivalency in order to get a valid answer.

I don't understand the point of this topic, and/or why it's so important. Seems common to me.

 

[Sage]

[font=Courier, Courier New, Monospace, monospace]0 = (1–1) + (1–1) + (1–1)
0 = 1 + (–1 + 1) + (–1 + 1) – 1[/font]

I think the key is that you iterate both equations the same number of times (an infinitely equal number of times? ). Notice that I have the same number of 1s in both equations, so both come out to 0.

 

[skip0110]

Whoops. I missed where he acknowledged that it was indeed invalid. But, still, you can't change groupings without maintaining equivalency in order to get a valid answer.

I don't understand the point of this topic, and/or why it's so important. Seems common to me.

When you find a sum like (1/2 -1/3) + (1/3 - 1/4) + (1/4 - 1/5) + ... they teach you to regroup and cancel, so the sum is 1/2 + (-1/3 + 1/3) + (-1/4 + 1/4) + ... = 1/2. Every calculus book I have seen teaches this technique.

So why does it not apply to my example?

Sage - In my above example, would I not have an extra tem at the end? But...ahh...I see it now...the last term is going to be 1/(something very big) so it is negligible! Thank you!

You may close this if it seems to not warrant further discussion. I get it now.

 

[Omnis]

[font=Courier, Courier New, Monospace, monospace]0 = (1–1) + (1–1) + (1–1)
0 = 1 + (–1 + 1) + (–1 + 1) – 1[/font]

I think the key is that you iterate both equations the same number of times (an infinitely equal number of times? ). Notice that I have the same number of 1s in both equations.

That makes sense. (kind of what I was trying to say before)

The grouping itself isn't valid without the -1 on the end, no? That's like totally unbalancing, and changing stuff.


Invalidly.

 

[Sage]

Uh, glad to have helped, if I did.

 

[VT]

[font=Courier, Courier New, Monospace, monospace]0 = (1–1) + (1–1) + (1–1)
0 = 1 + (–1 + 1) + (–1 + 1) – 1[/font]

I think the key is that you iterate both equations the same number of times (an infinitely equal number of times? ). Notice that I have the same number of 1s in both equations, so both come out to 0.

You are exactly right Sage. When Skip was regrouping he left out the last -1 which would cause it to equal 1 instead of 0. Its an error in the application of the associative property.