Convergence of $q_{1}+q_{1}^{2}+q_{2}^{3}+q_{1}^{4}+q_{2}^{5}+q_{3}^{6}+q_{1}^{7}+\ldots$

Question

I am trying to show that the series $$q_{1}+q_{1}^{2}+q_{2}^{3}+q_{1}^{4}+q_{2}^{5}+q_{3}^{6}+q_{1}^{7}+\ldots$$ where $q_{n}=q^{1+\left( \dfrac {4} {n}\right)} $, and $\left(0<q<1\right)$ is convergent, although the ratio of the $(n+1)$th term to the $n$th term is greater that unity, when n is not a triangular number. I am actually having a hard time recognizing the pattern in the series itself hence am struggling with, what the $(n+1)$th term to the $n$th really look like.

Answer 1

As Sivaram noted, $q_n\le q$ for all $n$. Thus the series converges because it is dominated by the geometric series for $q$, which converges.