I am facing a problem in which I know the equivalent of a term that I sum: $U_n(x) \sim \frac{A_n}{x^2}$. However, I am interested in the equivalent of my serie (for $x \rightarrow \infty$).
I tried "to see" to consider that I can just sum the equivalent but in my problem it gives me the good power but not the good coefficient : I find
$$\sum_{n\geq0} U_n(x) \sim \frac{\sum A_n}{x^2}$$
The $x^2$ is ok but the value $\sum A_n$ doesn't match my numeric estimation.
I know that if $a_n \sim b_n$ and $c_n \sim d_n$ then it is wrong in general to say $a_n+c_n \sim b_n + d_n$. This is why I have my problem here.