Writing improper integral and infinite sum with limit or not?

Question

I just wondering what is actually the correct notation of these two operators?

1) Writing the improper integral

Which one is correct?

a) $\displaystyle\int_0^{\infty}f(x) \mathbb dx$

b) $\displaystyle\lim_{b \to \infty}\displaystyle\int_0^{b}f(x) \mathbb dx$

2) Writing the infinite sum

Which one is correct?

a) $\displaystyle\sum_{k=0}^{\infty} b_k$

b) $\displaystyle\lim_{n \to \infty}\displaystyle\sum_{k=0}^{n} b_k$

c) $\displaystyle\sum_{k=0}^{n} b_k$

3) What about other Big Notation like Infinite Product, Infinite Union, Infinite Intersection, etc????

It's just notation, but i don't know that sometimes we have many argues how to write it correctly..

Answer 1

• In the context of the Riemann integral, we define the notation $\int_{a}^{\infty} f(x) \,\mathrm{d}x$ by the formula $$ \int_{a}^{\infty} f(x) \,\mathrm{d}x := \lim_{b\to\infty} \int_{a}^{b} f(x)\,\mathrm{d}x, $$ assuming that this limit on the right exists. Without such a definition, the notation $\int_{a}^{\infty} f(x) \,\mathrm{d}x$ is meaningless in the context of Riemann integration, since the integral is defined via Riemann sums on a compact (closed and bounded) interval.

• In the context of Lebesgue integration, the notation is defined slightly differently: $$ \int_{a}^{\infty} f(x) \,\mathrm{d}x := \int_{[a,\infty)} f(x)\,\mathrm{d}x. $$ The integral over the interval $[a,\infty)$ could probably be understood as a limit over integrals of the form $[a,b]$, but this is an unneccessary step in the context of Lebesgue integration.

• In the context of summation, the infinite series is defined in a limit (similar to how the Riemann integral is defined). Hence $$ \sum_{n=1}^{\infty} a_n := \lim_{N\to\infty} \sum_{n=1}^{N} a_n. $$ That is, the series is defined as the limit of the partial sums (assuming that this limit exists).

• Infinite products may be similarly defined by a limit, e.g. $$ \prod_{n=1}^{\infty} a_n := \lim_{N\to\infty} \prod_{n=1}^{N} a_n. $$ However there are some technicalities in dealing with infinite products. In particular, the notion of convergence is slightly different: for example, the product $$ \prod_{n=1}^{\infty} \frac{1}{n} $$ is said to "diverge to 0". This is because the usual approach to infinite products is to make use of the logarithm. That is, $$ \prod_{n=1}^{\infty} a_n = L \iff \sum_{n=1}^{\infty} \log(a_n) = \log(L). $$ Indeed, I believe that this is the typical definition of an infinite product.

• When dealing with integrals, series, and products, we have topological tools which allow us to make sense of limits. In the context of unions and intersections of sets, we typically don't have those tools (note: there are categorical tools which let us talk about limits (and colimits), but such tools are often too advanced for elementary students, and we don't need them to talk about infinite unions). Instead of using limits, infinite unions and intersections are defined by $$ \bigcup_{n=1}^{\infty} A_n := \{ a : \exists n \text{ s.t. } a \in A_n \} \qquad\text{and}\qquad \bigcap_{n=1}^{\infty} A_n := \{ a : \forall n, a \in A_n \} .$$ That is, the union is the set of elements which are contained in at least one of the sets in the union, and the intersection is the set of elements which are contained in all of the sets in the intersection.

Answer 2

$\sum_{k=0}^\infty b_k$ is exactly the definition of $\lim_{n\to\infty}\sum_{k=0}^n b_k$ when the limit exists. And $\int_0^\infty f(x)dx$ is the definition of $\lim_{b\to\infty}\int_0^b f(x)dx$ when $f$ is Riemann integrable in $[0,b]$ for all $b>0$ and the limit exists. So you can use any of these notations, but of course it is shorter to write it without limits.