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.

>>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?