Need an explanation on a certain terminology about converegence range in series.

Question

I found x's possible values, but I am also required to check the edges. What does it mean to check the edges? and why is it necessary for solving the problem?

Answer 1

If you compute the radius of convergence $r$ of a real power series with center $c$, then the power series converges for all $x$ in the open intervall with radius $r$ around $c$. But what remains to check is what happens at the two (so far excluded) endpoints of the intervall and that is what you need to check separately.

Answer 2

The limit test you used gives you the convergence radius $r$, which you calculated as $\sqrt[3]2$ (I didn't check the calculation).

What you then know is that the series converges for

$$-r < 2x+1 < r \Longleftrightarrow |2x+1| < r$$ and diverges for

$|2x+1| > r$.

What you don't yet know from the test is what happens when $|2x+1|=r$. Considering that, when considering real values only, the set of points where the series converges will always be an interval, it makes sense to call the check what happens when $2x+1=r$ and $2x+1=-r$ "checking the edges".

Those 2 points are the endpoints ("edges") of the interval $[-r,r]$, and you are asked to check what happens on those ednpoints: will the series converge or not? That is something the limit test can't tell you!