how can 2 = 0?

College Confidential Community
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<p>tanonev, that is not a valid point. Every term is included, with it's correct sign. You have, nonetheless, made an interesting observation of one of the riddles of infinity. It is indeed possible to partion all natural numbers into disjoint subsets, each with one odd and infinitely many even numbers, even though in some sense, there is an "equal number" of odd and even natural numbers.</p>

<p>As for why that grouping is not valid... that's going to keep me up at night. The issue lies in the fact that an infinite summation is not in fact a summation but a limit of partial summations, and you need to rigorously prove that any sort of re-ordering of terms converges to the same "sum," and apparently in this case it does not. It sure feels like it should, though. I suspect the reason is related to what tanonev observed. Even if every terms is "present," the even terms are in some sense counted "sooner" than the odd terms, thus every partial sum, and thus the limit thereof, is scewed down.</p>

<p>The best answer I can come up with is this: an infinite sum is a limit of partial sums. The sum, as towerpumpkin writes is, is not in this form, it is in the form of a limit of partial sums of a bunch of limits of partial sums, and one cannot automatically equate this to a different limit simply based on intuition. As far as I know, grouping terms in FINITE subsets is kosher. I'm fairly sure I can prove that with a little thinking. If anyone finds such a dissection that leads to a similar absurdity, let me know.</p>

<p>Thanks for posting that, towerpumpkin. If I ever teach a real analysis course, I will be sure to use that as an example of what one must always rigorously prove things that seem plausible; they may well be staggeringly false.</p>

<p>This is sort of similar to another interesting contradiction: consider the function f(x)=0 for x <= 0, f(x)=e^(-1/x) for x>0. It is fairly easy to see that the Taylor series centered at 0 converges to 0 for all x, so one might suggest e^(-1/x)=0 for all x>0, which is absurd. The problem is that while the series may converge, the Remainder term does not approach 0, which is the true criterion for Taylor series representations.</p>

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<p>1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ... = ln(2) is a conditionally convergent series, not an absolutely convergent series</p>

<p>Theorem (first proved by Riemann): if sum(a<em>n) is a conditionally convergent series and r is any real number whatsoever, then there is a rearrangement of sum(a</em>n) that has a sum equal to r</p>

<p>However, this does not imply that the limits themselves are equal, since infinite series cannot be treated in the same manner as finite series.</p>

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<p>tanoneV:</p>

<p>the term of any conditionally convergent series can be rearranged to give either a divergent series or a conditionally convergent seres whose sum is any given umber S. eg</p>

<p>ln2=1-1/2+1/3-1/4+1/5-1/6+...</p>

<p>rearrange it: (1-1/2-1/4)+(1/3-1/6-1/8)+(1/5-1/10-1/12)+... =1/2ln2</p>

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<p>"However, this does not imply that the limits themselves are equal, since infinite series cannot be treated in the same manner as finite series."
i don't understand this.</p>

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<p>I was referring to an earlier post that called the rearrangement thing a contradiction, while there is actually nothing wrong with it. It's just, er, wacky...</p>

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<p><a href="1-1/2-1/4...">quote</a>+(1/3-1/6-1/12...)+(1/5-1/10-1/20...)...=0+0+0...=0</p>

<p>Meaning you keep subtracting, like the first part would be (1-1/2-1/4-1/8-1/16-1/32...)

[/quote]
</p>

<p>not exactly. here, each term tends to zero, but is not zero itself. so each is kinda like 0.000000000000....0001. when you add up infinte terms like that, it makes a finite value.

[quote]
HiWei that is only assuming that it is sequential and not simultaneous</p>

<p>How about this:</p>

<p>∞=∞</p>

<p>(1+2+3+4...)=∞
2<em>(1+2+3+4....)=2</em>∞=(2+4+6+8...)
(2+4+6+8...)=∞
∞=2<em>∞
1</em>∞/∞=2*∞/∞
1=2</p>

<p>

[/quote]

dividing by infinty is invalid</p>

<p>
[quote]
What I never really got, though...
[IMPORTANT INFORMATION IN BINARY] XOR [RANDOM JUNK] = [MORE RANDOM JUNK]</p>

<p>and</p>

<p>[RANDOM JUNK] XOR [MORE RANDOM JUNK] = [IMPORTANT INFORMATION IN BINARY]</p>

<p>violation of the conservation of information???

[/quote]

is there a law for consrvation of information. i somehow dont think so.</p>

<p>and the first one is obvious. how come the second one comes about?</p>

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<p>are people here randomly looking up complex looking mathematical terms and names and fitting them in wherever possible or something?</p>

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<p>"not exactly. here, each term tends to zero, but is not zero itself. so each is kinda like 0.000000000000....0001. when you add up infinte terms like that, it makes a finite value."</p>

<p>Actually, the sum does equal zero. As I said before, since the series is conditionally convergent, you can rearrange it to (correctly) give any sum you please.
1 - 1/2 + 1/3 - 1/4 + ... = ln(2)
1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + ... = (3/2)ln(2)</p>

<p>"is there a law for consrvation of information. i somehow dont think so.</p>

<p>and the first one is obvious. how come the second one comes about?"</p>

<p>I think there is conservation of information...even through black holes (though I'm not sure what they decided in the end on this one...)</p>

<p>Anyhow, let me abbreviate IIIB, RJ, and MRJ for those three long all caps things.
[IIIB] XOR [RJ] = <a href="we%20agreed%20about%20that%20one">MRJ</a>
[IIIB] XOR [IIIB] XOR [RJ] = [IIIB] XOR <a href="same%20thing%20to%20both%20sides">MRJ</a>
0 XOR [RJ] = [IIIB] XOR <a href="%5BX%5D%20XOR%20%5BX%5D%20=%200">MRJ</a>
[RJ] = [IIIB] XOR <a href="0%20XOR%20%5BX%5D%20=%20%5BX%5D">MRJ</a>
[RJ] XOR [MRJ] = [IIIB] XOR [MRJ] XOR <a href="same%20thing%20to%20both%20sides">MRJ</a>
[RJ] XOR [MRJ] = [IIIB] XOR 0
[RJ] XOR [MRJ] = [IIIB], QED</p>

<p>"are people here randomly looking up complex looking mathematical terms and names and fitting them in wherever possible or something?"
"conditionally convergent" is a Calc BC term
"conservation of information" is probably not a real term, but the concept exists, I'm pretty sure...I think it follows from conservation of mass/energy, since information must be stored in at least one of the two.</p>

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<p>
[quote]
Actually, the sum does equal zero. As I said before, since the series is conditionally convergent, you can rearrange it to (correctly) give any sum you please.

[/quote]
</p>

<p>um....</p>

<p>and i never said people were making up terms. i said they were randomly using terms which they didnt understand.</p>

<p>
[quote]
Actually, the sum does equal zero

[/quote]

i beg to differ.</p>

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<p><a href="http://www.math.unl.edu/%7Ewebnotes/classes/class48/prp730.htm%5B/url%5D"&gt;http://www.math.unl.edu/~webnotes/classes/class48/prp730.htm&lt;/a&gt;&lt;/p>

<p>Since 0 is a real number, there exists <em>some</em> rearrangement of the alternating harmonic series that "adds up" to 0, and there's no reason (besides our instinctive unease when dealing with infinity) that (1-1/2-1/4...)+(1/3-1/6-1/12...)+(1/5-1/10-1/20...)... is not that rearrangement. If, however, you want to be pedantic and have it fit, rearrange as follows:
1 - 1/2 - 1/4 - 1/8 -... --> 0+ (the infinitely small positive number)
... - 1/6 = a little more than -1/6
... + 1/3 = a little more than 1/6
... - 1/12 - 1/24 - 1/48 - ... --> 0+ (the infinitely small positive number)
... - 1/10 = a little more than -1/10
... + 1/5 = a little more than 1/10</p>

<p>so the partial sums of (1 - 1/2 - 1/4 - ...) + (-1/6 + 1/3 - 1/12 - 1/24 - ...) + ... taken by parenthetical groups oscillate around 0 and get closer and closer to 0, implying that their limit is indeed 0. (And the "infinitely small positive number" I referred to does NOT get larger with each step, because there are an infinite number of terms in each "set" of negative terms, so we can make that number as small as we please.)</p>

<p>And I know very well what "conditionally convergent" means. Don't automatically assume that we don't know what a term means just because we don't define it. We're assuming that the rest of the people here know what it means.</p>

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<p>The International Whaling Commission (IWC) was set up by the International Convention for the Regulation of Whaling on December 2, 1946 with a headquarters in Cambridge, England. The role of the inter-governmental commission is to periodically review and revise the Schedule to the Convention, controlling the conduct of whaling by setting the protection of certain species; designating areas as whale sanctuaries; setting limits on the numbers and size of catches; prescribing open and closed seasons and areas for whaling; controlling aboriginal subsistence whaling; and other measures.</p>

<p>Each signatory state of the convention is represented by a Commissioner at the IWC. There are currently 45 members. The IWC has three main committees - Scientific, Technical, and Finance and Administration. Meetings are held annually in May or June and are generally extremely divisive - demonstrating a complete split on all major issues between the pro-whaling nations and their supporters and the anti-whaling nations.</p>

<p>The IWC introduced an open ended moratorium on all commercial whaling in 1986. However the Convention grants special permits to allow whale killing for scientific purposes. Since 1986 only Norway, Iceland and especially Japan have been issued with permits, with Japan being the sole permit holder since 1995 as part of their 16-year programme. Norway lodged a protest to the zero catch limits in 1992 and is not bound by them.</p>

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<p>Um...what? Why is there a post about whaling?</p>

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<p>I wish I could understand this type of math. Oh well, this further proves my deficiency. I thought Alg 2H would improve my math abilities or at least give me some brownie points, LoL. I guess not.</p>