Is there a way to compute an asymptotic formula of $\sum_{n\leq x}\lceil 2\sqrt{n} \rceil$?

Question

I was trying to find an asymptotic formula for the sum $$ \sum_{n\leq x}\lceil 2\sqrt{n} \rceil$$ using Euler–Maclaurin summation formula.. but I really don't know how to proceed in this case, mainly because of the ceil function ($\lceil x \rceil$ is the smallest integer greater than or equal to x). Thank you in advance, every hint/solution will be much appreciated!

Answer 1

Adding to The Phoenix's excellent comment on exact formulas: using summation by parts (by whatever name), we know that $$ \sum_{n \leq x} 2 \sqrt{n} = \frac{4}{3} x^{3/2} + O(x^{1/2}). \label{1}\tag{1} $$ (For reference this calculation is in e.g. Theorem 3.2 of Apostol's Introduction to Analytic Number Theory.)

This sum cannot differ all that much from the sum $$ \sum_{n \leq x} \lceil 2 \sqrt{n} \rceil $$ we are interested in, because $0 \leq \lceil t \rceil - t < 1$. Hence, the worst case scenario for each of the approximately $x$ terms in the sum is that they contribute an error compared to $\eqref{1}$ of size $1$. Therefore $$ \sum_{n \leq x} \lceil 2 \sqrt{n} \rceil = \frac{4}{3} x^{3/2} + O(x). $$