Not absolutely convergent series

All you faggots talking about 0.999~ = 1.

That's nowhere near the hardest thing to get your head round.

Take the alternating harmonic series.

1 - 1/2 + 1/3 - 1/4 .......
 
It's pretty simple to show it equals ln(2)

But if I re-arrange the order of the terms I can make it equal whatever the hell I want.

That's a mindfuck.

So for example

1 + 1/3 - 1/2 + 1/5 + 1/ - 1/4 ..... != 1 - 1/2 + 1/3 - 1/4 ...

even though the individual terms in them are exactly the same.

Prove it faggot.

>>1
1 - 1/2 + 1/3 - 1/4 ...
This is a logical progression where the ``...'' is obvious.

1 + 1/3 - 1/2 + 1/5 + 1/ - 1/4 ...
This is not.

it's not a mindfuck, because you're bouncing the convergence differently by changing the order of the bounces, thus you can achieve different convergence values by doing that.

>>4
i don't know, when i first learned this it did blow me over a bit
i was like, whoa

This is a result of Analysis that I enjoyed immensely. It's pretty counter-intuitive, at least initially, that you can get it to equal ANYTHING you want.

>>4
Intuitively though, the order shouldn't matter, because you're summing all of them. Like with that fucking hotel, infinity is a bitch sometimes.

>>7
You have a bad intuition. You should intuitively think of a limit of a sum as a restriction on summing.

what the fuck are you fuckers talking abourt

looks like a bunch of bullshit

>>3

OP

Obvious progression, two odd, one even.

>>2

It's a simple proof. Consider it as two series of positive and negative terms. Both these series must be divergent, otherwise the first series is absolutely convergent.

Then just define your series so that the first terms are from the positive series and add to more than your limit, then the next terms are negative until your series totals below the limit you want.

>>8
Do they characterize the sum as a "limit of a sum" or as the sum being the same as a limit?  Do they ever use the phrase "limit of a sum" if the texts define sums themselves to be limits?  That is, do we say 1) sums have limits (implying that they do not really achieve that value themselves) or 2) sums are limits (implying that they achieve that value)?  That is, is limiting something applied to a set of terms attempting a sum, or is limiting something describing a sum?

>>12
The value of an infinite sum is defined as the limit of its partial sums. I think you're confusing yourself a little with semantics.

>>7
Is that the "aleph hotel" that was used to describe the properties and orders of infinity?