Timeline for Non-absolute convergence of series with asymtotically equal coefficients

9 events

when toggle format	what		by	license	comment
May 15, 2010 at 2:35	vote	accept	wood
May 14, 2010 at 7:57	answer	added	Emerton		timeline score: 2
May 13, 2010 at 22:42	history	edited	Pete L. Clark	CC BY-SA 2.5	added 8 characters in body
May 13, 2010 at 19:28	answer	added	Julián Aguirre		timeline score: 2
May 13, 2010 at 17:52	comment	added	Theo Johnson-Freyd		Emerton's comment basically does it, I think --- you need to be able to control either the signs of the $\delta_n$ s or their sizes --- if $\sum \delta_n$ converges absolutely, for example, then you're golden, and this requires only that the approximation $f(n) \approx g(n)$ gets sufficiently better as $n\to \infty$. Emerton: you should leave your comment as an answer.
May 13, 2010 at 14:15	comment	added	Emerton		Just subtracting one series from the other, it seems that you need $\sum_{n = 0}^{\infty} (f(n)-g(n))$ to converge. (This is what fails in Xandi Tuni's example.) Writing $f(n)/g(n) = 1 + \delta_n,$ so that $\sum_{n = 0}^{\infty} (f(n) - g(n)) = \sum_{n = 0}^{\infty} \delta_n g(n),$ you see need control over the signs of the $\delta_n$. E.g. if they are all of the same sign, you are okay, while if the sign of $\delta_n$ is always the same as, or always opposite to, that of $\delta_n, then you are in bad shape. (This is what goes wrong in Xandi Tuni's example.)
May 13, 2010 at 13:10	answer	added	Xandi Tuni		timeline score: 12
May 13, 2010 at 12:47	history	edited	wood		edited tags
May 13, 2010 at 12:41	history	asked	wood	CC BY-SA 2.5