# Thread: What is the mistake?

1. x = 1+2+4+8+16+32+....

x-1 = 2+4+8+16+32+....

x-1 = 2(1+2+4+8+16+32+....)

1+2+4+8+16+32+.... = x

x-1 = 2x x = -1

What's wrong? What is your opinion?

2.

3. The first thing is your sequence contains a '6' so the '...' is meaningless.
Your 3rd line also has a 6, which should be a 3.
If fact I think the whole thing is a bit of a mess.... Sorry..

4. i agree with the dead guy

5. Originally Posted by infrared
x = 1+2+4+6+8+16+32+....

x-1 = 2+4+6+8+16+32+....

x-1 = 2(1+2+4+6+8+16+32+....)
In Europe we call it the "two times table". In your series x -1 you have no 3, so you may not have 6 in your x - 1 = 2(blah blah...) series. Likewise you have no 12 in 2x.

As billco rightly said.

6. remove the 6, add the general term to make it more obvious what this series is:

x = 1+2+4+8+16+32+....+2^n+....

x-1 = 2+4+8+16+32+....

x-1 = 2(1+2+4+8+16+32+....)

1+2+4+8+16+32+.... = x

x-1 = 2x

x = -1

Now what's wrong?

7. um, the first series is divergent - which is bad thing in general and unforgivable in this case

8. Sorry, you are right, 6 isn't in this series. I corrected it. In this case what is the answer?

9. Originally Posted by infrared
x = 1+2+4+8+16+32+....

x-1 = 2+4+8+16+32+....

x-1 = 2(1+2+4+8+16+32+....)

1+2+4+8+16+32+.... = x

x-1 = 2x x = -1

What's wrong? What is your opinion?
Well now what you have is lines 1 & 4 are equal so 2 & 3 are meaningless. and then line 5 is... well just line 5. Have we found enough errors for you ?

10. This isn't as silly as it might seem at first sight.

Although I don't do numbers that well, as such, I think I see it. Take Shmoe's hint and this one here:

<generator, index>

11. river_rat explained what's wrong with it, it's a divergent series, so you can't expect any of the usual operations you can perform on convergent sereis to be valid in any way.

Now, what's right about it? Is there a good reason to assign the divergent sum 1+2+4+8+... the value -1? There is actually. What the heck, does anyone want to try to come up with an explanation of why it could be considered sensible to say 1+2+4+8+...=-1? (though a mildly sloppy abuse of the '=' sign here).

As a hint to what I'm getting at, I declare it's also reasonable to assign the sum 1+3+9+27+... the value of -1/2.

12. That is what Euler liked to do schmoe - its an example of a continuation on the real line if i am getting my jargon correct.

Easy part - 1 + a + a^2 + a^3 + ... + a^n + ... = 1/(1-a) if |a| < 1

However, the function f(a) = 1\(1-a) makes sense on a larger region then just the open interval (-1, 1)! in fact it makes sense on R \ {1} - and so we can extend our function to almost the entire real line (though the original idea of it being a geometric series only holds in (-1, 1))

Extend this idea (with some other fancy stuff like multivalued functions) to the complex plane gives you the powerful idea of analytic continuation.

13. Yep, that'd be the one. The complex plane puts everything together so nicely. That series is defined for all complex values of a with |a|, the 1/(1-a) version gives an analytic continuation to the complex plane ecept a pole at a=1. That analytic continuations are unique where they exist (so not the case in the real version) makes this the choice for extending the definition of this sum to other values.

It is an abuse of the "=" sign to say 1/(1-2)=1+2+4+8+... but analytic continuation like this is just one of several useful ways to assign finite values to otherwise divergent sums (Hardy's Divergent Series text is a place to find more). Methods for divergent sums are handy in some areas physics I'm told, techniques going by the term "Zeta regularization".

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