Why is it that the interval of convergence is half open? I am given the following power series and asked to find the radius of convergence and determine the exact interval of convergence
$$\sum\biggr(\frac{3^n}{n\cdot 4^{n}}\bigg)x^n \Leftrightarrow \sum\bigg(\frac{3}{n^{1/n}\cdot 4}\bigg)^{n}x^n$$
If $a_n=(\frac{3}{n^{1/n}\cdot 4})^{n}$, then $\lim\sup|a_n|^{1/n}=\frac{3}{4}$, hence, the radius of convergence is $R=\frac{4}{3}$.
I'm then given that the interval of convergence is $I=[-\frac{4}{3},\frac{4}{3})$.
I am confused how exactly the author determined that the I.O.C. is half open, and how I can determine if the I.O.C. will be open, closed or half open?

Series convergence tests, e.g., Hadamard's formula for checking radius of convergence, are often inconclusive along the boundary; you must check the boundary separately, by plugging in boundary values, thus obtaining a new series (with no x's involved anymore) and again using some series convergence test, to check for convergence.  
