Question #41338

This question tests the summation of a series involving step functions where the general term cannot be simplified using standard algebraic limits or telescoping methods.

The trap lies in assuming that the addition of a vanishing term $1/\sqrt{n}$ to $\sqrt{n}$ only affects the greatest integer function at exactly the boundary points $n = (k+1)^2 - 1$, leading to an incorrect count of 1 jump per interval.

Students often fail to partition the domain into intervals $[k^2, (k+1)^2 - 1]$ and rigorously solve the inequality $\sqrt{n} + 1/\sqrt{n} \ge k+1$.

By doing so, they miss that the function jumps for the last two integers of every interval for $k \ge 2$, and all three for $k=1$, rendering 'General Term' approximations invalid.