This question is a weird one. I may have a potential answer for it, but I'm not confident in how good it is.

Discuss the convergence of the series (as completely as possible)

$\displaystyle \left[\left(1+\frac{1}{2}\right)^2-\left(1+\frac{1}{3}\right)^2\right]+\left[\left(1+\frac{1}{4}\right)^2-\left(1+\frac{1}{5}\right)^2\right]+...+\left[\left(1+\frac{1}{2n}\right)^2-\left(1+\frac{1}{2n+1}\right)^2\right]+...$

What is the effect of removing the square brackets?

My current answer is uncertain, but here's what I have.

If the series is $\displaystyle \left(1+\frac{1}{2n}\right)^2$, we have

$\displaystyle \lim_{n\rightarrow\infty}\left(1+\frac{1}{2n}\righ t)^2=1$

At a certain point, $\displaystyle a_n=\left(1+\frac{1}{2n}\right)^2$ is nearly equal to 1.

As such, adding an infinite number of 1's together cannot be a finite number.

Therefore, the series is divergent.

Does my answer have any holes in its logic? And I also haven't figured out what the effect of "removing the square brackets" would do.