Quote:
Originally Posted by reuben_tate
Zapmeister...you can't use (1-x)^(-2) at x=1 as an analytic continuation of 1+2x+3x^2+... because (1-x)^(-2) itself is not analytic at x=1.
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crap... you got me. you got me damn good.
best i can say now is that if you try to define sums of divergent series using the formulae and stuff you derive from convergent series, you have to throw away a lot of favourable nice assumptions you make about how they behave.
e.g.
sum(n: n=1 to infinity) = x (here x is assumed to be -1/12 but that doesn't matter)
=> sum(n: n=0 to infinity) = x
=> sum(n-1: n=1 to infinity) = x
=> sum(n: n=0 to infinity) - sum(n-1: n=1 to infinity) = 0
=> sum(1: n=1 to infinity) = 0
=> sum(1: n=0 to infinity) = 0
=> sum(1: n=0 to infinity) - sum(1: n=1 to infinity) = 0
=> 1=0
however i'm pretty sure term-by-term addition/subtraction like this is not justified in the way stargroup100 is trying to treat these things, although he never explicitly states that in any of his posts.
i'm still trying to find a contradiction that i'm satisfied with but for the moment it looks like you got me here >_<
edit: i was hoping my 300th post would be less self-demeaning than this but whatever