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I would like to know the relationships between bounded and convergent series. By bounded series I mean a series whose sequence of partial sums is bounded. For example, it seems natural that if a series is convergent, it is also bounded, but does the converse hold?

Thanks in advance,

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    @Federico: There is no need to "sign" your message: Your name/user name already appears on the bottom right of the message automatically.2011-09-07

3 Answers 3

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Whenever we have a series, $\sum_{i=1}^{\infty} a_i,$ we "automatically" get two sequences out of that series:

  1. The sequence of terms, which is $a_1,a_2,a_3,\ldots$; and
  2. The sequence of partial sums, which is $s_1,s_2,s_3,\ldots$, where $\begin{align*} s_1 &= a_1\\ s_2 &= a_1+a_2\\ s_3 &= a_1+a_2+a_3\\ &\vdots\\ s_n &= \sum_{i=1}^n a_i = a_1+a_2+\cdots + a_n. \end{align*}$

When we talk about "convergence of the series", we are really talking about convergence of the sequence of partial sums: the series $\sum a_i$ converges if and only if the sequence $(s_n)$ converges. That is, your definitions about "series" are really about "sequence of partial sums", and so you have the usual relationship:

In particular, $\sum_{i=1}^{\infty}a_i\text{ converges}\Longleftrightarrow \{s_i\}_{i=1}^{\infty}\text{ converges}\Longrightarrow \{s_i\}_{i=1}^{\infty}\text{ is bounded}\Longleftrightarrow \sum_{i=1}^{\infty}a_i\text{ is bounded}$ (where "is bounded" is as per your definition above); but it is possible for $\{s_i\}_{i=1}^{\infty}$ to be bounded, and not convergent, so one can have a series $\sum_{i=1}^{\infty}a_i$ that is bounded (i.e., the sequence of partial sums is bounded) but does not converge.

A simple example of this is $\sum_{i=1}^{\infty} (-1)^n$. The partial sums are $s_{2k+1} = -1$ and $s_{2k}=0$ for every $k$, so the sequence of partial sums is: $-1,\ 0,\ -1,\ 0,\ -1,\ldots$ which is bounded but not convergent. So the series is bounded but not convergent.

The relevant theorem for sequences, as you are no doubt aware, is:

Theorem. If $\{b_n\}$ is a monotone sequence, then $\{b_n\}$ converges if and only if it is bounded.

How does that translate for series? When is the sequence of partial sums monotone?

$\{s_i\}$ is increasing if and only if $s_n\leq s_{n+1}$ for all $n$, if and only if $s_{n+1}-s_n\geq 0$ for all $n$; but $s_{n+1}-s_n = a_{n+1}$. So:

The sequence of partial sums of $\displaystyle \sum_{i=1}^{\infty}a_i$ is increasing if and only if all the terms $a_i$ are nonnegative. The sequence of partials sums is strictly increasing if and only if all the terms $a_i$ are positive.

Likewise,

The sequence of partial sums of $\displaystyle \sum_{i=1}^{\infty}a_i$ is decreasing if and only if all the terms $a_i$ are nonpositive. The sequence of partial sums is strictly decreasing if and only if all the terms $a_i$ are negative.

So we conclude:

Theorem. Let $\displaystyle \sum_{i=1}^{\infty}a_i$ is a series in which every term $a_i$ is nonnegative. Then the series converges if and only if it is bounded (in the sense that the sequence of partial sums is bounded).

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    You said it is possible that one can have a series that is bounded (i.e., the sequence of partial sums is bounded) but does not converge. Is this also true for the absolutely convergent case? So is it possible to have a sequence of partial sums that is bounded, but where the associated series does not converge absolutely?2016-07-21
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No, a bounded series does not necessarily converge. Consider the series $\displaystyle \sum (-1)^n $ (heavily related to Henning's example). It will forever oscillate between 0 and 1 (or -1 and 0, depending on the indices).

But if the partial sums are bounded and monotonic, then it does converge.

But in either case, it's a bit weaker than the converse - convergent series always have bounded partial sums.

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    @Didier: I think that is a far stronger condition, and I way attempting to ask for a 'surprising' or 'deceptive' set of conditions. Perhaps I was not clear with my parenthetical additions? Anyhow, thank you for your help.2011-09-07
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A convergent sequence is bounded, but a bounded sequence is not necessarily convergent. Consider, for example the sequence (1, -1, 1, -1, 1, -1, ...).

On the other hand, an increasing (or decreasing) bounded sequence in $\mathbb R$ will necessarily converge.

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    @Federico: if all terms of the series are positive, then the sequence of partial sums is increasing; if all terms of the series are negative, then the sequence of partial sums is decreasing.2011-09-06