Talk:Convergent series

message: The statement of the Cauchy criterion for convergence is incorrect, but I am not sure
how it should be changed.
David Radcliffe 19:35, 30 November 2005 (UTC)
replies: text: I think it is correct just fine. :) It says that the sequence of partial sums is a Cauchy sequence, which is equivalent to the sequence being convergent. Oleg Alexandrov (talk) 00:20, 1 December 2005 (UTC)

text: You are right, it was close to Cauchy sequences, but not quite. The harmonic series would have satisfied that criterion which was wrong.  I also posted a formulation after you fixed that part. Oleg Alexandrov (talk) 03:10, 1 December 2005 (UTC)
thread_title: Mistake?

message: Unde the convergence tests section, the article could also mention the n-th term test, the p-series, the geometric series, and most importantly, the power series. If anyone willing to add these tests, please do so. Otherwise I'll add them in later on.--Rejnal 04:33, 15 March 2006 (UTC)
replies: text: Yes, please add them. If you feel you have a lot of detail to put in, you could also create new articles about those topics. Cheers, Oleg Alexandrov (talk) 04:37, 15 March 2006 (UTC)
thread_title:  More convergence tests 

message: I just skimmed the article and did not see any mention of real numbers or complex numbers or anything.  So, do all of the tests work for complex numbers?  I am pretty sure not all work for complex numbers, but many do if written appropriately.  So, here are a few suggestions.  The first 3 are basically the same idea, the 4th is different.
1. Mention that we are talking about complex and real numbers here.
2. For each test, or each part of the article, be specific about which of the two types of numbers it works for.
3. Make things more general.  The ratio test, for example, as stated only works for series which are all positive real numbers.  But, in fact it works for series with positive and negative terms and even complex series.  It should be stated more generally, I think, so that it is more useful.
4. An even more general section could be added which talks about convergent series in a general topological space.  I believe this article is probably mostly accessed by high school and college students who do not know a whole lot about that, so it might be best to put this new section near the end so as to not scare off people who are just trying to learn about basic series for real numbers.  But, it seems fitting for it to at least be mentioned.  Maybe there is another article that could be referenced at least.  Or, don't add this section and entitle this article "Convergent series of real and complex numbers".
StatisticsMan (talk) 17:33, 28 February 2009 (UTC)
replies: text: The discussion here is all real. No section can be added about convergent sequences in topological spaces because in general there is no such thing. Sequences can converge (consider neighbourhoods around points), but in general there is no operation of summation, which is clearly needed for series. Clearing up the complex cases a bit is a good idea though.— Kan8eDie (talk) 14:04, 2 March 2009 (UTC)
thread_title:  Real numbers, complex numbers?  A few suggestions I do not have time to work on myself 

message: I think the equality signs should be changed to -> or <  or > as the case may be.
Obviously the series never takes the value so it can never be an equality.
So the questions "find the sum of the series.." can be treated as flawed in most cases can it not?  —Preceding unsigned comment added by 192.18.192.76 (talk) 07:21, 20 July 2009 (UTC)
thread_title:  equality 

message: Why in the world does Convergence (mathematics) redirect here?! Series convergence is just a special case of sequence convergence, which is a special case of convergence of nets, which can further be generalized using filters. In other words, the notion of convergence is much more general than convergence of series, so there really should be a separate page called Convergence (mathematics) that discusses the general concept. Actually, it might make more sense for Convergence (mathematics) to just redirect to Limit (mathematics) (instead of Convergent series) since the concepts of "convergence" and "limit" are essentially the same, but I don't know how to do redirects. 216.160.76.121 (talk) 03:19, 30 August 2009 (UTC)
replies: text: I agree, I redirected to limit (mathematics)--Work permit (talk) 05:18, 30 August 2009 (UTC)

text: The Riemann zeta function is defined by analytic continuation of the series Σ n−s. Bo Jacoby (talk) 18:58, 30 July 2010 (UTC)
thread_title: More general notions of convergence?!

message: 1. What is the sum of the reciprocals of the super squares?
*: {1 \over 1^1}+{1 \over 2^2}+{1 \over 3^3}+{1 \over 4^4}+{1 \over 5^5}+{1 \over 6^6}+\cdots = ?.
2. What is the sum of the reciprocals of the super powers of two?
*: {1 \over 2}+{1 \over 2^2}+{1 \over 2^{2^2}}+{1 \over 2^{2^{2^2}}}+{1 \over 2^{2^{2^{2^2}}}}+{1 \over 2^{2^{2^{2^{2^2}}}}}+\cdots = ?.
3. What is the sum of the reciprocals of the Partion numbers?
*: {1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 5}+{1 \over 7}+{1 \over 11}+\cdots = ?.
4. What is the sum of the reciprocals of the Ulam numbers?
*: {1 \over 1}+{1 \over 2}+{1 \over 3}+{1 \over 4}+{1 \over 6}+{1 \over 8}+\cdots = ?.
5. What is the sum of the reciprocals of the Superfactorials?
*: {1 \over 1!}+{1 \over 1! \times 2!}+{1 \over 1! \times 2! \times 3!}+{1 \over 1! \times 2! \times 3! \times 4!}+{1 \over 1! \times  2! \times 3! \times 4! \times 5!}+{1 \over 1! \times 2! \times 3! \times 4! \times 5! \times 6!}+\cdots = ?.
6. What is the sum of the reciprocals of the triangular Factorial numbers?
 
*: {1 \over 1!}+{1 \over 1!+2!}+{1 \over 1!+2!+3!}+{1 \over 1!+2!+3!+4!}+{1 \over 1!+2!+3!+4!+5!}+{1 \over 1!+2!+3!+4!+5!+6!}+\cdots = ?.
I'm looking for closed forms. Robo37 (talk) 10:51, 15 August 2011 (UTC)
thread_title:  Six convergent series needing solving. 

message: How did Euler prove the sum of the reciprocals of the primes diverges? I know he created the Riemann zeta function for it but how did he do this?  — Preceding unsigned comment added by 86.185.99.78 (talk) 21:18, 11 December 2012 (UTC)
thread_title:  Reciprocals of primes 

message: The following sentence is misleading:
The power series of the logarithm is conditionally convergent.
The power series of log(1+z) around 0 is absolutely convergent for |z|<1, divergent for |z|>1 and for |z|=1 various things can happen (the radius of convergence is 1).
Sprocedato (talk) 11:26, 21 January 2013 (UTC)
thread_title:  Conditionally convergent