I need to investigate convergence and absolute convergence of series $$\sum_{n=1}^{+\infty} \frac{(2x)^n}{x^2n+\frac{70}{n}}$$ depending on the value of the parameter $x \in \mathbb{R}$
I think use here Root test $$\lim_{n \to +\infty} \sqrt[n]{\left| \frac{(2x)^n}{x^2n+\frac{70}{n}}\right| } = 2x $$
Okey, for $x >\frac{1}{2}$ series is divergent and for $x <\frac{1}{2}$ is converges absolutly, but what about $x = \frac{1}{2}$ ?
So for $x = \frac{1}{n}$ should be $$\sum_{n=1}^{+\infty} \frac{1}{\frac{n}{4} + \frac{70}{n}} $$ I think it is diverges by the comparison test. If I take $\sum_{n=1}^{+\infty} \frac{1}{n}$ it will be right?
$$\sum_{n=1}^{+\infty} \frac{1}{\frac{n}{4} + \frac{70}{n}}>\sum_{n=18}^{+\infty} \frac{1}{\frac{n}{4} + \frac{70}{n}}>\sum_{n=18}^{+\infty} \frac{1}{\frac{n}{2}}=\infty$$so this series is divergent for $x=\dfrac{1}{2}$