/sci/ - Science & Math

>>10881789
Bit late, hope you still see this

>For example, can I take every other number out and create a new divergent series from that and still preserve information?
No. Consider the sequence

1 + 1 + 1 + 1/4 + 1 + 1/9 + 1 + 1/16 + ...

The odd index numbers are just 1, the even index numbers are reciprocals of squares. Obviously 1 + 1 + 1 + ... diverges, and 1 + 1/4 + 1/9 + ... is pi^2/6.


> What about re-aranging terms in the series?
https://en.wikipedia.org/wiki/Riemann_series_theorem


>Is addition in a divergent series commutative? What about associativity, for example:
Not really, consider these two (same?) sequences

1 + (1 - 1) + (1 - 1) + (1 - 1) + ... = 1 + 0 + 0 + ... = 1
(1 + 1) - (1 + 1) - (1 + 1) - (1 + 1) - ... = 2 - 2 - 2 - 2 - 2 - 2 ... which diverges

Not sure if this is what you're asking for though, because you wanted a divergent series, and this series isn't really divergent or convergent. Also subtraction isn't commutative in the first place, so perhaps this was a bad example. But in general, you can't just sum infinite series in a different order and expect the same result.

>>10882967
Don't use Baby Rudin if it's your first time doing Analysis. I recommend Tao's books, they're free and easily accessible to starting math students. One complaint I have about Rudin is that many of his solutions to his exercises come out of nowhere and are unrelated to the chapter, I think it would only confuse beginners.


>there are people on /mg/ who can't solve pic related