Talk:Real Analysis/Series

I think there's a little problem here. A series can't really be the limit of the sequence of its partial sums... For example, we call $$\sum_{n=0}^{+\infty} (-1)^n$$ a series, but the limit of its partial sums, which are $$s_n:=\frac{1+(-1)^n}{2}$$, does't even exist.

I'd like to suggest some solutions for the problem pointed out; I hope my contri could be useful (even if my english isn't very good).

Formally, a series with summands $$(a_n)$$ is an ordered pair of sequences, namely $$((a_n),(s_n))$$, the second of which can be obtained by the first with the following rule:

(I) $$s_n:=\begin{cases} a_0 & \text{, if } n=0\\ s_{n-1}+a_n & \text{, if } n\ge 1 \end{cases}$$

The series $$((a_n),(s_n))$$ could be denoted by a simbol like $$\sum a_n$$; then $$(s_n)$$ can be called sequence of the partial sums of $$\sum a_n$$.

If $$(s_n)$$ is regular (i.e. exists $$\lim_{n\to \infty} s_n$$) then the number $$\sum_{n=0}^{+\infty} a_n:=\lim_{n\to +\infty} s_n$$ is called sum of $$\sum a_n$$ and the series $$\sum a_n$$ is said to be regular; if $$(s_n)$$ isn't regular, we say that $$\sum a_n$$ is not regular; if $$(s_n)$$ is convergent, we say that $$\sum a_n$$ is convergent and so on...

Note that rule (I) is actually invertible: infact, if we choose a sequence $$(b_n)$$ there is only one sequence $$(a_n)$$ s.t. $$(s_n)=(b_n)$$, that is:

(II) $$a_n:=\begin{cases} b_0 & \text{, if } n=0\\ b_n-b_{n-1} & \text{, if } n\ge 1 \end{cases}$$

This fact shows that a series is univocally determined not only by its summands, but also by its partial sums; in other words, we have only one degree of freedom when we choose the couple $$((a_n),(s_n))$$ to form a series.

Your work is good; keep writing about Mathematical Analysis!

Bye, Gugo82.


 * Thank you for pointing this out. I was wrestling over which way to go with this.  But in the end your right that is standard to call $$\sum_{n=0}^{+\infty} (-1)^n$$ a series.  I didn't want to spend a long time on definitions of formal sums, because at the level this book is currently written I think it distracts from getting into the subject.  But I may come back and change this later.  I first would like to go through the book and get everything correct, and roughly in its usual state of affairs, and then think about the structure of the book and what should be added.  Thanks again, and always feel free to be bold and edit the page directly.  Thenub314 (talk) 19:35, 5 February 2009 (UTC)