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I try to understand something I've read:

it is sometimes possible to assign a value to a series, called the sum of the series. This value is the limit as $n$ tends to infinity (if the limit exists) of the finite sums of the $n$ first terms of the series, which are called the $n$th partial sums of the series. That is, $$\sum_{i=1}^\infty a_i = \lim_{n\to\infty} \sum_{i=1}^n a_i.$$ When this limit exists, one says that the series is convergent or summable, or that the sequence $(a_1,a_2,a_3,\ldots)$ is summable. In this case, the limit is called the sum of the series. Otherwise, the series is said to be divergent.

So my basic question is:

When a sequence is summable is that also convergent?

Does this mean that there is a connection between a series and a sequence, if you prove a series is convergent, the sequence is also?

For the sake of simplicity make $a\in\mathbb{Z^+}$ and $n,m\in\mathbb{N}$.

A term in this sequence is connected to the previous term: $a_n,a_{n+1},..a_m$. But it is unknown wether the last terms of the sequence converges.

If we show that $a_n+a_{n+1}+..a_m$ is convergent (has a limit) when $m\rightarrow\infty$, does it mean that the sequence converges? What about if $m$ is set to a finite number?

Basically, can one prove something about the sequence by showing its series are convergent?

What are connections between series and sequences?

Does it have anything to say if the series are geometric and arithmetic, for the sequence to converge or diverge?

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  • $\begingroup$ You can easily prove that if $\sum |a_n| < \infty$, then $a_n \rightarrow 0$. $\endgroup$ – dcolazin Jul 18 at 14:53
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    $\begingroup$ Please do write down your definition of "summable series"" ...as opposed, apparently, to "convergent series". $\endgroup$ – DonAntonio Jul 18 at 14:53
  • $\begingroup$ In order that a series be summable, the sequence must converge to 0. $\endgroup$ – user247327 Jul 18 at 14:54
  • $\begingroup$ Sequences aren't summed, series are. $\endgroup$ – Randall Jul 18 at 14:54
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    $\begingroup$ OK, I included that content from Wikipedia. $\endgroup$ – Matthew Leingang Jul 18 at 15:04
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I don't think there's a very definite question up there, but I understand why it might have been hard for you to pinpoint exactly what was troubling you, or to even express it efficiently supposing you could pinpoint it.

So what I'll just do is answer the titular question (by explaining the link between sequences and series), and then I'll talk about summability.

First of all, a ramble about terminology. The word sequence in modern mathematics is often exclusively reserved for a succession of objects -- thus a sequence of steps, a sequence of notes, a sequence of operations, a sequence of numbers, a sequence of sequences, and so on. However, both in nonmathematical speech and in ancient mathematics, this word was used more or less synonymously with the word series, which in those contexts mean quite the same thing. So we have a series of tones, a series of blasts, etc. However, in contemporary mathematics the word series is often exclusively used to refer to a particular type of sequence that has become very ubiquitous in mathematics -- a sequence of additions. Thus by a series is meant a sequence of addition operations. For example, start with any real number you like, and transform it by adding to it any real number you like (positive or negative or zero), then another to the sum, then another to the new sum, and so on, and you have what is called a series. What I have actually done is explain in words what's meant by the (quite informal) symbol $$a_1+a_2+a_3+a_4+a_5+a_6+\cdots,$$ which is more formally written as $$\sum_{i\ge 1}{a_i}.$$ That is, a series is just a way of representing a sequence of cumulative sums, each sum in the $i$th position in the sequence consisting of $i$ summands. This sequence is what's called the sequence of partial sums of the series.

In this way you see the link between series and sequences; every series is actually a sequence of sums as described above; on the flip side every sequence can be thought of as some series, namely the series obtained by summing the differences between consecutive terms of the sequence to its first term.

Finally, we can talk about infinite series; that simply means that the addition does not end. This question arises because such series do not always give a number as a result even if the terms are all numbers. They exhibit different behaviours; some series simply accumulate to become larger than any number; some don't even have a predictable behavior, changing course every now and then. The definition given is to pin down those series of numbers that -- though infinite -- add up to a real number. These are the ones called convergent, or summable.

The definition simply asks you to consider the sequence of partial sums of a series, and to see if this sequence converges (becomes stable and settles to a real number); if so, it says that such series will also be called convergent, or summable.

In sum, it's all about sequences. A series is just a sequence of additions, or a sequence of partial sums; conversely, every sequence of addable objects (in particular, numbers), can be decomposed into a series.

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  • $\begingroup$ To obtain a sequence $b_n$ such that its series is $a_n$, we do $b_n:=a_n-a_{n-1}$. Also, the symbol $\sum_{i\ge1}a_i$ usually rather means an explicit number (the sum of the series) and not the sequence of partial sums. $\endgroup$ – Berci Jul 18 at 17:21
  • $\begingroup$ @Berci The symbol $\sum a_i$ is used to signify a series; as explained above, a series is just a sequence, so the symbol is a sequence. In some cases, it may represent the sum of the series itself, but this is not the primary signification since not all series have sums, yet we represent them all by that symbol. $\endgroup$ – Allawonder Jul 18 at 21:33
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I think you're getting very confused about all the various terminology around, so I'll first try to provide these definitions, and then try to illustrate any links between the concepts.

Definition $1$. A sequence in $\Bbb{R}$ is a function $a: \Bbb{N} \to \Bbb{R}$.

That's all there is in the definition of a sequence. Normally, the value of the function $a$ at a point $n$ in its domain is denoted by the symbol $a(n)$. However, it is more common in the context of sequences to write $a_n$ instead of $a(n)$. As a result, we often speak of "the sequence $(a_n)_{n=1}^{\infty}$."

Definition $2$. Let $a: \Bbb{N} \to \Bbb{R}$ be a given sequence, and define a new sequence $s: \Bbb{N} \to \Bbb{R}$ by the rule \begin{align} s_n = \sum_{i=1}^n a_i. \end{align} We define the sequence $a$ to be summable if the sequence $s$ has a limit; i.e if $\lim \limits_{n \to \infty} s_n$ exists. If this limit exists, we denote it by the symbol $ \sum_ \limits{i=1}^\infty a_i, $ and we call it the sum of the sequence $a$.

The sequence $s$ is often called the "sequence of partial sums of $a$". Strictly speaking, those are the only two definitions you need to know. Now, using only this terminology, we have the following theorem:

Theorem: If $a: \Bbb{N} \to \Bbb{R}$ is a summable sequence, then $\lim \limits_{n \to \infty} a_n$ exists and equals $0$.

However, often, rather than saying "the sequence $a$ is summable", we often use the terminology "the series $\sum_ \limits{i=1}^\infty a_i$ converges" (I find this to be pretty peculiar terminology, but sometimes you just have to accept that there are several terminology around to describe the same thing; some are more precise, and others are more convenient/traditional).

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Note that $\sum_{n=0}^\infty a_n$ converges$\implies \lim_{n\rightarrow \infty}a_n=0$

i.e. convergence or summability of series $\sum_{n=0}^\infty a_n$ necessitates convergence of sequence $<a_n>$ to the limit point $0$.

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