Talk:Conditional convergence

spectral sequence

Conditional convergence of a Spectral sequence is a big issue and should be mentioned or even be explained on this page. The best reference is probably Boardman's famous article Conditionally convergent spectral sequences. 84.62.195.98 (talk) 19:53, 28 January 2012 (UTC)

defintion

The definition on top of this web page currently says: «In mathematics, a series or integral is said to be conditionally convergent if it converges, but it does not converge absolutely.» I have some problems with this definition. Rudin's book is the only reference in this page. In 3.46 Rudin says: «If $\scriptstyle \sum \limits _{n=0}^{\infty }a_{n}$ converges, but $\scriptstyle \sum \limits _{n=0}^{\infty }|a_{n}|$ diverges, we say that $\scriptstyle \sum \limits _{n=0}^{\infty }a_{n}$ converges non-absolutely.» I glanced around, I could not find anywhere the above definition in Rudin's book. Another problem. I would imagine that conditional convergence be the opposite of Unconditional convergence; but this is not the case. I then searched around. I found Conditional Convergence in Math World that reports a definition similar to the above; but it cites the book An introduction to the theory of infinite series and (searching in the above link) that book never uses the word conditional or unconditional. This is really weird. I think that this page should be investigated and possibly changed. Mennucc (talk) 16:58, 15 May 2018 (UTC)

I see that as a problem, too. The article on Unconditional convergence explicitly states that conditional convergence is the exact opposite of unconditional convergence. That is, the given series converges, but there exist reorderings of the series which diverge or converge to a different limit. The corresponding article on conditional convergence in the German wikipedia goes in the same line. Conditional convergence ist not always equivalent to non-absolute convergence, e.g. in infinite-dimensional Banach spaces. Gzim75 (talk) 11:38, 20 July 2023 (UTC)