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So I want to prove whether the following series converges absolutely or not:

$$\frac{1}{2}\sum_{n=0}^\infty (n^2+3n+2)q^n$$

where $ q \in \mathbb{C}, \mid q\mid<1.$

My attempt was:

$$\frac{1}{2}\sum_{n=0}^\infty \mid(n^2+3n+2)q^n\mid = \frac{1}{2}\sum_{n=0}^\infty \mid(n^2+3n+2)\mid \mid q^n\mid$$

It holds that $ n \geq 0 \Rightarrow n^2+3n+2 > 0 \,\, \forall n$ therefore,

$$\Rightarrow \frac{1}{2}\sum_{n=0}^\infty \mid(n^2+3n+2)\mid \mid q^n\mid = \sum_{n=0}^\infty (n^2+3n+2)\mid q^n\mid $$

For showing that $\frac{1}{2}\sum_{n=0}^\infty (n^2+3n+2)\mid q^n\mid$ converges it is sufficient to show that $\sum_{n=0}^\infty (n^2+3n+2)\mid q^n\mid$ converges.

To be continued...

My problem is now that I don't really know which convergence criterion I should use nor I know what to do with that $\mid q^n \mid$ part whether I can omit the absolute value or not (my guess is not since it is a complex number and $\mid .\mid \;: \mathbb{C} \rightarrow \mathbb{R}$ and that basically everything.

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