Why do some people use $+\infty$ instead of $\infty$?

Why do some people use $+\infty$ instead of $\infty$? Because we usually denote minus infinity as $-\infty$, I think it is sufficient to denote plus infinity as $\infty$. Are there any reasons for this notation? I saw that many people majoring analysis use like $\int_0^{+\infty}$ or $\sum_{j=1}^{+\infty}$ or $\bigcup_{j=1}^{+\infty}$ which is not related to the unsigned infinity.

Solution 1:

One reason one might want to do this is to distinguish between positive infinity ($+\infty$) and unsigned infinity ($\infty$).

Solution 2:

ADD Regarding the more informal $+\infty$ versus $\infty$ in sums and integrals, using $+\infty$ rather than $\infty$ might be more seen since we can encounter ourselves with $$\int_{-\infty}^{+\infty}$$ situations, as well as

$$\sum_{-\infty}^{+\infty}$$

or

$$\bigcup_{n=1}^{+\infty}$$

where we want to make the disctinction of signs.

I personally prefer, though I don't use much for convention reasons, the alternative

$$\int_{\Bbb R}$$

$$\sum_{n\in \Bbb Z}$$

and

$$\bigcup_{n\in \Bbb N}$$

There is a slight difference between $+\infty$ and $\infty$. In the context of the extended reals, we talk about two points, $+\infty$ and $-\infty$, which make $\Bbb R$ look like $[-1,1]$. We add two points at the end so that every sequence $\{x_n\}$ has a limit point in $\Bbb R^*$. A limit point of a sequence $\{x_n\}$ can be defined as the limit of a subsequence of $\{x_n\}$: intuitively, this limit point will have points of $\{x_n\}$ cluttering closer and closer to it. As an example, the sequence $(-1)^n\left(1+\frac 1 n\right)$ has both $-1$ and $1$ as limit points.

If the sequence is bounded, then we get a limmit point (a convergent subsequence, equivalently), by Bolzano Weiertrass, and if the sequence is unbounded we get $+\infty$ or $-\infty$ as a limit point (take as an example $0,1,-1,2,-2,3,\dots$, which will have both of them). We make a disctinction between them to get this particular construction. Note that when we talk about $+\infty$ and $-\infty$ we talk about order:

$$-\infty < x < +\infty$$ for every real $x$.

But we can as well adjoint only one point, $\infty$, and then $\Bbb R$ will rather look like a closed circle: we're "joining" the real line by its "endpoints", and again we get a compact space: every unbounded sequence, regardless of sign, has $\infty$ as a limit point.

When we're dealing with $\Bbb C$, since there is no order, we only talk about $\infty$, which we "indentify" as the boundary of the complex plane. Intuitively, we're taking the plane, which basically looks like a disk without the boundary, and closing it off in the following manner: we take a rope around the circle and pull through both ends, getting a sphere, just like we close those bags that have that rope going inside them. Basically, we now get that every sequence in $\Bbb C$ has a limit point: if it is bounded, we get one by Bolzano Weiertrass, and if it is not, we get that $\infty$ is a limit point.

Solution 3:

When dealing with the one point compactification of the real numbers (where you associate the whole number line with a circle minus one point, and define the missing point to be infinity), there is only one infinity, so it is unsigned.

However when dealing with the extended real line, you have $+\infty$ and $-\infty$ defined, and I guess it's helpful to know, just by looking at the symbol used, which notion of infinity you are using.