Writing improper integral and infinite sum with limit or not? 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.. 

$\sum_{k=0}^\infty b_k$ 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. 
