Consider the sequence $$ x_n=\frac{1}{n^{b}}\left(\frac{1}{n^{a}}+1\right)^{\frac{1}{c}}.$$ where $a,c>0,b\in \mathbb{R}.$
I think the series $\sum\limits_{n\in \mathbb{N}} x_n$ converges for $b>1,a>0,c>0$ as $ x_n=\frac{1}{n^{b}}\left(\frac{1}{n^{a}}+1\right)^{\frac{1}{c}} \sim\frac{1}{n^{b}}\left(\frac{1}{cn^{a}}+1\right) \sim \frac{1}{n^b}$ and diverges for $b\leq 1$
Is it correct?