I'm having some trouble making the connection between infinite series and the underlying sequences that they are summed over and both of their connections to functions. This question is a bit long, but please bear with me.

Wikipedia quote: A series is informally speaking the sum of the terms of an infinite sequence


A series is defined as the limit as $n \to \infty$ of it's partial sums $S_n$

My Understanding

If we want to define a series $\sum a_n$, the only way to do it, is to take the limit as $n \to \infty$ of it's partial sums $S_n$.

It's partial sums, being the finite summations from $i = 0$, to $n$ of a base infinite sequence $\{a_n\}_{n=0}^{\infty}$.

This base infinite sequence $\{a_n\}$ is really just a list of the infinite outputs of some function $f$. We can say that there exists a function $f : \mathbb{Z^+} \to \mathbb{R} $ such that $f(n) = a_n$, so essentially what we are really doing when we say ' we are taking the sum of the terms of an infinite sequence' is summing over the integer outputs of the function $f$.

So $\sum a_n$ is really just $\sum f(n)$, where $f : \mathbb{Z^+} \to \mathbb{R} $

It's partial sums $\sum_{i=0}^{n} a_i$, are sums with respect to the base infinite sequence $\{a_n\}_{n=0}^{\infty}$. Now if we think of this in terms of functions, the partial sum would be the sum of integer outputs of the function $f$ over an interval $[0, n]$, whereas the series would the sum of integer outputs of the function $f$ on the interval $[0, \infty)$

Partial Sum : $$\sum_{i \ \in \ [0, n]} f(n) \ \ \text{where}\ f : \mathbb{Z^+} \to \mathbb{R}$$

Series : $$\sum_{i \ \in \ [0, \infty)} f(n) \ \ \text{where}\ f : \mathbb{Z^+} \to \mathbb{R}$$

Now, the partial sums then in turn, form a finite sequence, $S_0, S_1, ... S_n$, which can be expressed as $\{S_n\}_{i=1}^{n}$.

If we had to write it out $\{S_n\} = \{a_0, (a_0 + a_1), (a_0 + a_1 + a_2), ... , (a_0 + a_1 +a_2 + ... + a_n)\}$

This sequence created by the partial sums is again really just a list of finite outputs of another function $g$. Again we can say that there exists a function $g : \mathbb{Z^+} \to \mathbb{R} $ such that $g(n) = S_n$. But $g$ is different from $f$ in that $g$ is a function whose outputs give the partial sums $S_n$, whereas the outputs of $f$ give the elements of the original sequence. And as it turns out $g$ is of more importance conceptually than $f$, because we determine convergence/divergence of $\sum f(n)$ from $g$.

Now by taking the limit of our sequence of partial sums, which is taking the limit of our function $g$ as $n \to \infty$, what we are doing is finding the value the function $g$ (the sums of our original base sequence $\{a_n\}$) converges towards (assuming the limit exists), and from that we can determine the sum of our series.

Essentially to make the jump from a finite sum to a series (from a finite sum to an an infinite one), all we do is evaluate a set of finite sums of increasing length with respect to a base infinite sequence (which is really just a function) and see if it tends towards some finite value (converges) (or not -diverges)

If the limit exists and is finite, we say the series converges, and the value of the limit is the value of the series, and if the limit is infinite or doesn't exist, then we say the series diverges, and the value of the series is $+\infty$ or $-\infty$

Everything I've said above can be compactly written below as follows:

$$\begin{align} \sum_{n=0}^{\infty}a_n &\stackrel{\text{def}}{=} \lim_{n \ \to \infty} S_n \\ &= \lim_{n \ \to \ \infty} \underbrace{\sum_{i=0}^{n} a_i}_\text{Partial Sum} \\ \\ &= \underbrace{\lim_{n \to \infty} \left\{\gamma n \right \}_{i=0}^{n}}_\text{* (1)} \\ \\ &\text{And since} \ \exists \ g : \mathbb{Z^+} \to \mathbb{R} \ni g : n \mapsto \gamma n \\ \\ &= \lim_{n \ \to \ \infty} \gamma n \end{align}$$

Note: In $*(1)$, $\gamma n$ denotes a closed form expression of $\sum_{i=0}^{n} a_i$ (the partial sum). In this step we take the limit of the sequence of partial sums, which is as I mentioned above is essentially taking the limit of $g(n)$ as $n \to \infty$

Firstly if there is anything you spot wrong with my understanding of sequences and series, please inform me, as there is nothing I want more than to rectify misconceptions I might have.

Secondly from the point of view of analysis is my reasoning and understanding backwards or circular?

Is it fine for me to interpret sequences and series as I've done, are sequences really just the list of function outputs in disguise or are they a whole different beast completely?

I ask this as I've read the following quote :

The two basic concepts of calculus, differentiation and integration, are defined in terms of limits (Newton quotients and Riemann sums). In addition to these is a third fundamental limit process: infinite series.

And it got me wondering, are sequences and series more fundamental than functions themselves or is it the other way around? Which comes first from the point of view of analysis, sequences and series or functions?

  • $\begingroup$ Something that I forgot to add in this post, which would've made it too long, is that interpreting the sequences and series as I did, as functions, makes the results that the sequence ${\frac{1}{x}}$ converges yet the series $\sum_{i=0}^{\infty}\frac{1}{x}$ diverges extremely explicit as $f$ in my interpretation above can converge and $g$ need not converge. $\endgroup$ – Perturbative Aug 21 '16 at 14:28

You can define a function $f:X\to Y$ from and to any sets $X$ and $Y$, provided only that they are not empty.

As you have said, a sequence of real numbers is a function from $\Bbb N$ to $\Bbb R$. You can define sequences of anything (not only of real numbers), but as a function, the domain of a sequence must be $\Bbb N$, so sequences are particular cases of functions.

To define limits you need a topology. If you don't know what a topology is, the idea is that you need a way to define expressions like "for big enough $n$" or "tends to".

To define series you need a sum and the above mentioned topology. Although you don't need the explicit concept of topology in $\Bbb R$ to compute limits, it's still there.

What I mean is that the functions are very basic objects. Sequences are special functions and series are a complex (i. e., complicated) construct that need a rich structure to be defined.

Nevertheless, it must be noted that in some textbooks about calculus, some elementary functions are defined using series. But elementary functions are a very special kind of functions.

  • $\begingroup$ Thanks for this answer. I felt that the way sequences and series were treated in most introductory calculus (not analysis) courses to be very hand wavy, but what you've said has put some concrete basis for sequences and series . Thank you for that! $\endgroup$ – Perturbative Aug 21 '16 at 14:12

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