Deadly Question - Very Very deadly. (1 Viewer)

[KeypadSDM]

Now let's just assume that:

Ln[2] = 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... To infinity

(We could prove this, but it'd just take too long for me to be bothered.)

Now rearranging:

Ln[2] = 1 + 1/3 + 1/5 + 1/7 + ... - (1/2 + 1/4 + 1/6 + ...)

Adding cancelling terms:

Ln[2] = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... - (1/2 + 1/4 + 1/6 + ...) - (1/2 + 1/4 + 1/6 + ...)

Ln[2] = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... -2(1/2 + 1/4 + 1/6 + ...)

Ln[2] = 1 + 1/2 + 1/3 + 1/4 + 1/5 + ... - 1 - 1/2 - 1/3 - 1/4 - 1/5 - ...

Ln[2] = 0

So where's the error?


Edit: 1 Hour later and no one is game to attempt. Maybe you're all at parties at 11 o'clock on a saturday night.

 

[wogboy]

I seem to remember a theorem stating that if you rearrange the order of a conditionally convergent sequence (which is not absolutely convergent), the sequence will converge to a different sum.

For example if you have the conditionally (but not absolutely) convergent series S = s1 + s2 + s3 + ... , where every term will alternate between positive to negative in sign. If you group all the positive terms together and add them together, you will get +infinity. Likewise if you just pick out the negative terms and add them together you will get -infinity. So what you can do is pick just positive terms and add them together until you only just exceed a particular number say L. Then you can pick negative terms and add them on until you get something just less than L. Then add positive numbers until you just exceed L, and repeat this process forever. Then sum will converge to L as you continue this process for an infinite length of time. You can see that L is completely arbitrary (up to you, L can equal 0, 1, 2000, -6 etc). This means you can rearrange such a series so that it converges to a different number.

All that you've done is rearrange the series:
1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ..., which converges to L=ln[2], into a different order such that L=0.

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In case anybody's wondering what conditonally convergent & absolutely convergent means:

* Conditionally convergent: This is when a sum of numbers is finite (i.e. not infinite). A series S = {s1, s2, ...} is conditionally convergent if (s1 + s2 + s3 ....) is finite.

* Absolutely convergent: This is when a sum of numbers is finite (i.e. conditionally convergent) AND when the sum of the absolute values of the numbers is finite too. A series S = {s1, s2, ...} is absolutely convergent when:

i) s1 + s2 + s3 + ..... is finite AND;
ii) |s1| + |s2| + |s3| + .... is finite.

For example 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... is absolutely convergent, but 1 - 1/2 + 1/4 - 1/6 + ... is only conditionally convergent. And 1 + 2 + 3 + 4 + 5 + ... is not convergent at all!

Bah I'm out of here, off to bed, 'nite.

 

[turtle_2468]

So, just to restate wogboy's argument, it is possible to arrange any conditionally convergent sequence to sum to any term you like.

 

[KeypadSDM]

Well, that's a much more mathematical approach to answering it. When I first thought of it I did something like this:

Let A = 1 + 1/3 + 1/5 + 1/7 + 1/9 + ...
B = 1/2 + 1/4 + 1/6 + 1/8 + ...

.: Ln[2] = A - B

Note, 2 * B = B + A
:. B = A

:. Ln[2] = A - A
= 0

But to disprove this i said that in the equation

2 * B = B + A

Had effectively 1 lot of infinity on the left hand side, while it had 2 lots of infinity on the right hand side.

So when you subtracted B from either side, you effectively had no infinites on the left and one infinity on the right. This makes no sense at all and makes the relationship A = B meaningless. Or at least in my opinion.