I was reading the Proof to Theorem 3.54 in Rudin's Prininples of Mathematical Analysis. The theorem says the following:

Let $\sum a_n$ be a series of real numbers which converges but not absolutely. Suppose $-\infty\leq\alpha\leq\beta\leq\infty$. Then there exists a rearrangement $\sum a_n'$ with partial sums $s_n'$ such that $\liminf s_n'=\alpha$ and $\limsup s_n'=\beta$.

At some point in the proof, the author says "Let $P_1, P_2,\ldots$ denote the nonnegative terms of $\sum a_n$, in the order they occur"

I'm not sure what he means by this. Any ideas?

  • 1
    $\begingroup$ Can you share us a bit more context? At least, what is the theorem being proved? Anyway, most probably $P_1$ is the first nonnegative $a_{n_1}$, $\,P_2$ is the second nonnegative $a_{n_2}$ (with $n_1<n_2$ and all other $a_k<0$ for $n_1\ne k<n_2$) and so on.. $\endgroup$ – Berci Jan 29 '14 at 0:06
  • $\begingroup$ I probably won't be able to provide sufficient context but it's in page 77 in the link below math.boun.edu.tr/instructors/ozturk/eskiders/guz12m331/rud.pdf $\endgroup$ – mathemagician Jan 29 '14 at 0:13
  • $\begingroup$ If we could have something like the text of the theorem, that would make this question much more helpful as an archive of mathematical q/a, rather than only helpful to those who have Rudin. (Just a thought...) $\endgroup$ – apnorton Jan 29 '14 at 4:20

In my 1964 edition of Rudin, it's Theorem 3.55 (not 3.54). The series $\sum a_n$ is a "nonabsolutely convergent series of real numbers," which is only possible when infinitely many terms are positive and infinitely many are negative. The sequence $P_1,P_2,P_3,\ldots$ picks out the nonnegative terms. (Rudin also says "let $Q_1,Q_2,Q_3,\ldots$ be the absolute values of the negative terms of $\sum a_n$, also in their proper order.")

For example, if $\sum a_n = \sum_{n=1}^\infty{(-1)^n\over n}$, then the sequence of $P$'s is ${1\over2},{1\over4},{1\over6},\ldots$ and the $Q$'s are $1,{1\over3},{1\over5}\ldots.$

  • $\begingroup$ thanks, I think nonnegative terms of $\sum a_n$ is confusing. Why not say non-negative terms of $a_n$? $\endgroup$ – mathemagician Jan 29 '14 at 0:48
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    $\begingroup$ The terms of $\sum a_n$ are $a_n$. I don't know what are the terms of $a_n$. $\endgroup$ – GEdgar Jan 29 '14 at 1:08
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    $\begingroup$ @GEdgar, Rudin himself is a little logically lax in notation and terminology. On page 51, he introduces $\sum_{n=1}^\infty a_n$ (for which $\sum a_n$ is an abbreviation) as a symbolic expression for the sequence of partial sums $\{s_n\}$, with $s_n=a_1+\cdots+a_n$, but then immediately also uses it to denote the limit of that sequence (when the limit exists). In short, he uses the same expression for both a function defined on the positive integers (which is what a sequence is defined to be on page 23) and for a number. $\endgroup$ – Barry Cipra Jan 29 '14 at 1:44
  • $\begingroup$ @GEdgar $a_n$ is a sequence and it's common to talk about terms of a sequence. $\endgroup$ – mathemagician Jan 29 '14 at 4:44
  • $\begingroup$ @mathemagician, it's more common to think of $a_n$ as denoting a generic term in a sequence, and to denote the sequence itself by $\{a_n\}$ or $a_1,a_2,a_3,\ldots$. When what's of interest is the series $\sum a_n$, it's standard practice to refer to the things being added up as terms of the series. (Side note: On page 23, Rudin formally defines the term terms in his definition of sequence, but he never formally defines it for series. To the extent that an infinite series is a sequence of partial sums, the "terms" of a series are those sums, not the numbers that constitute them!) $\endgroup$ – Barry Cipra Jan 29 '14 at 12:18

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