However, we can still proceed the general investigation without need to worry about that subtlety and see what happens. Suppose we have a dominant convergent series change of variable:

$$u=\sum_{k=1}^{\infty}a_kx^k$$

Then since it is convergent, we can interchange the limits and hence derivatives and sums to get:

$$du=\sum_{k=1}^{\infty}a_kkx^{k-1}dx$$

So a simple looking integral such as:

$$\int u du = \int \sum_{l=1}^{\infty}a_lx^l\left(\sum_{k=1}^{\infty}a_kkx^{k-1}\right) dx= \int \sum_{l=1}^{\infty}\sum_{k=1}^{\infty}a_la_kx^{l+k-1}dx$$

[Unrelated, continuation of previous posts] The above preliminary investigation suggest we can generate some infinite series representations that involve dominant convergence,from some functions and their integrals.

But can we push this further by starting with the constant function, integrate it, made a series change of variables, and then somehow remove the variable x that is initially introduced...?

Suppose we have a number of interest, e.g. $\frac{\pi}{2}$, we want to somehow lock it inside an integral with it, i.e. $\exists f(x)$

This explains why, combined with my impatience, I get quite sensitive, and espeicallty I have a real life friend I sever contact with last year because I cannot handle his increasingly help vampire behaviour despite he has very interesting and unorthodox viewpoints, which the two tings combined result in me having an unstable psyche for that year as I struggle to attend his problems and hoping that some of his unique insights I can learnt from,
(but this is getting unprofitable as his help vampire behaviour intensified since his electromagentism course starts