I'm trying to figure out the general term for a series where you subtract from every perfect square number, its square root.
So
$0-0+1-1+2+3+4-2+5+6+7+8+9+10+11+12+13+14+15+16-4+17+18+19+20+21+22+23+24+25-5+26+\cdots$
Is what's baffling me. I want to determine whether this is convergent or divergent, and I'm not sure how to do it without having a general term handy.
You could work with
$$\lim_{n\to\infty}\sum_{k=1}^{n^2}k-\sum_{k=1}^n k=\frac{(n^2+1)n^2}{2}-\frac{(n+1)n}{2}.$$