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/sci/ - Science & Math

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File: 21 KB, 579x149, lelelele.jpg [View same] [iqdb] [saucenao] [google]
6847019 No.6847019 [Reply] [Original]

hi /sci/

this is supposed to be wrong, why?
I suppose it's something similar as to why

SUM ( n) = - 1/12, n = 1,2,3,...

but I can't find a sound explaination.

>> No.6847023

>>6847019
if a series doesn't converge absolutely you can't do tricks like this.

The absolute values of top line are the harmonic series which doesn't converge.

>> No.6847694

>>6847023
This

>> No.6847734

>>6847019
You can't add divergent series.

Actually, you can say that you have the right to do it, the result being the demonstration of Bernett's Identity.

>> No.6847736

>>6847023
>tricks like this
>adding

okay.

>> No.6847738

>>6847023
This
http://en.wikipedia.org/wiki/Conditional_convergence

>> No.6847741 [DELETED] 

>>6847736
>fuding around with an infinite series without checking whether you can actually do this
>not a trick
okay

>> No.6847744
File: 2.79 MB, 853x480, infinite decimals.webm [View same] [iqdb] [saucenao] [google]
6847744

>>6847736
Adding is an operation on two numbers. You are adding infinite numbers.

>> No.6847751

>>6847744
I feel we are going to get into semantics, but what OP posted is adding two numbers, although he does it an infinite amount of times. Which is different from adding infinite numbers.

>> No.6847760

>>6847751
The trick OP is forbidden from applying is rearranging the infinite sum on the third line.

>> No.6847764

>>6847760
So infinite sums don't obey commutivity? For what reason?

Not trolling, would appreciate a link that describes why it fails in detail.

>> No.6847779

>>6847764
Not him but because the limit of a series is defined as the limit of the partial sums. Shuffling around the order can change the value of the partial sums and thus of the limit of the series.

>> No.6847790

>>6847779
Why is the limit defined through partial sums?

>> No.6847791

>>6847744
sauce on the video?

>> No.6847807

>>6847764
Well, there was the Wikipedia article already. And OP already provided a counterexample.

But it's not that hard to see intuitively. You have to remember that the infinite sum is by definition the limit of the partial sums. You can make the positive part of the partial sum bigger by rearranging the order so that the positive terms get added earlier. If the positive and negative parts of the infinite sum are finite on their own, then the amount by which you can advantage the positive part of the partial sum by is limited and will decrease to zero. But if the positive and negate parts are individually infinite, you can continue giving the positive part an advantage indefinitely, and that advantage need not approach zero.

>> No.6847811

>>6847791
It's from Chuunibyou. It's an edit, they were originally talking about magic.

>> No.6847817

>>6847811
too bad, i was actually exited about a show talking about constructive maths.

>> No.6847876

http://en.wikipedia.org/wiki/Riemann_series_theorem

Yeah so basically any conditionally convergent series can just be rearranged to equal whatever number you feel like making it equal. When I show you something like

<span class="math">1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdot \cdot \cdot = \ln(2)[/spoiler]

you can rearrange the terms like so,

<span class="math">1+\frac{1}{3}-\frac{1}{2}+\frac{1}{5}+\frac{1}{7}-\cdot \cdot \cdot = \frac{3}{2}\ln(2)[/spoiler]

the result OP got. OP, your math is entirely correct, but the rearrangement theorem is what's giving you this counter-intuitive result.

>> No.6848005

>>6847790
You got another rigorous way to define infinite sums, shitlord?
And don't tell me you're gonna do some retarded shit like performing arithmetic infinitely many times as in OP's picture.