Does the syntax $\sum_{n \geq 1}$ mean the same thing as $\sum_{n = 1}^{\infty}$?

Question

Is the syntax $\sum_{n \geq 1}$ equivalent to $\sum_{n = 1}^{\infty}$? Is it just another way of writing the same thing?

Answer 1

The quick answer is: it depends. It might mean the same thing, but some people would see a difference, which can be important in some situations.

About a given $(u_n)_{n \in \mathbb{N}^*}$, a sequence of real numbers, one can say different things:

• the sequence ("of partial sums") $\left(\sum^n_{i=1} u_i\right)_{n \in \mathbb{N}}$ is convergent; one usually says that the series (associated to the sequence) converges;
• the value of the series is some real number (i.e. the limit of the above sequence of partial sums);
• the sequence is summable and the value of the sum is some real number (to be defined below).

For pedagogical purposes, it might be worthwhile to forbid students to write things whose existence has not been previously shown. For example, some teachers don't like things like

$\lim_{n\to \infty} \frac{\sqrt{n+1}}{\sqrt{n}} = \lim_{n\to \infty} \sqrt{1+\frac{1}{n}} = 1$ so the sequence $(\frac{\sqrt{n+1}}{\sqrt{n}})_{n \in \mathbb{N}}$ converges

because one shouldn't write $\lim_{n\to \infty} \frac{\sqrt{n+1}}{\sqrt{n}} = \cdots$ before having proved that the sequence is convergent.

Something like

the series $\sum^{+\infty}_{n=1} u_n$ is convergent and $\sum^{+\infty}_{n=1} u_n = \frac{\pi^2}{6}$"

would also be weird, since the same notation $\sum^{+\infty}_{n=1} u_n$ is used to denote two different things in the same sentence: the series as an object in itself and its value. In other words, if we replace $\sum^{+\infty}_{n=1} u_n$ by $\frac{\pi^2}{6}$ in the above sentence, we get

the series $\frac{\pi^2}{6}$ is convergent

which is definitely weird.

For this reason, one of my teachers used to prefer the notation $\sum_{n\geq 1}$ to denote the series in itself, and $\sum^{\infty}_{n=1}$ to denote the value of this series, once it has been proved to be convergent. The $\sum^{\infty}_{n=1}$ notation for the value of a series, is, I think, quite universal; but I'm not sure about the notation $\sum_{n\geq 1}$.

The story could end here without further worries if you haven't heard of summable families or Lebesgue integration... But if you have, there is something else that can be denoted in a similar way and could produce another confusion.

Let $(u_n)_{n \in \mathbb{N}^*}$ be a sequence of, let's say, real numbers. The notation $\sum_{n \in \mathbb{N}^*} u_n$ refers to (if it exists) the only real number $S$ such that for every $\epsilon >0$, there exists $J \subset \mathbb{N}^*$ finite such that for every finite subset $J'$ of $\mathbb{N}^*$ containing $J$, we have $\vert \sum_{n \in J'} u_n - S \vert < \epsilon$.

If $\sum_{n \in \mathbb{N}^*} u_n$ exists, the sequence $(u_n)_{n \in \mathbb{N}^*}$ is said to be summable. Like S.K. mentioned it in his/her answer, this is equivalent to say that the function $ u : \mathbb{N}^* \rightarrow \mathbb{R}$ is Lebesgue-integrable, with respect to the counting measure.

One can, in fact, show that a sequence of real numbers is summable if and only the series associated to it is absolutely convergent.

So, for a convergent but not absolutely convergent series, $\sum^{+\infty}_{n=1} u_n$ exists (it is the value of the series) but $\sum_{n \in \mathbb{N}^*} u_n$ does not.

All that said, my advice would be: if you are a student, ask the teacher to tell you exactly what notation choices he/she makes and, when writing an exam, give precisions on what your notations refer to.

Answer 2

$\sum_{n\geq 1} a_n$ and $\sum_{n=1}^\infty a_n$ coincide if $(a_n)$ is absolutely convergent. The first symbol means "integration w.r.t. the counting measure", i.e. is defined as $$ \sum_{n\geq 1} a_n = \sup_{S \subset \mathbb{N}, |S| < \infty } \sum_{s \in S} \max(a_s,0) - \sup_{S \subset \mathbb{N}, |S| < \infty } \sum_{s \in S} \max(-a_s,0) $$ while the second symbol means $$ \sum_{n=1}^\infty a_n = \lim_{N\rightarrow\infty} \sum_{n=1}^N a_n. $$

The first symbol is not defined for some series, e.g. $a_n=\frac{(-1)^n}{n}$, because its positive and negative parts may both (!) be separately divergent, so we would get $\infty-\infty$. The advantage here is that the notation can be extended to arbitrary sets, even uncountable ones, e.g. $\sum_{x \in \mathbb{R}} f(x)$ makes sense for some $f$, namely those for which there are either at most countably many negative or positive terms and whose sum converges.

On the other hand, we have $\sum_{n=1}^\infty a_n = -\log(2)$. The important point is that both definitions agree for absolutely convergent sequences. In this case the summation order is immaterial. The first definition has the advantage, that one can use Fubini's theorem to justify the exchange of summation order, etc.

EDIT: The above convention for $\sum_{n\geq 1}$ is non-standard and depends on the field/context in question.