Example 1. Find the domain of convergence of the series

$$
\sum_{n=1}^{\infty} \frac{n^{3}}{\left(n^{2}+\sqrt{n}+1\right)^{x+1}}
$$
## Solution.

1. For each fixed $x$ all terms of the given series are positive:

$$
a_{n}=\frac{n^{3}}{\left(n^{2}+\sqrt{n}+1\right)^{x+1}}>0 \quad \forall n \geq 1
$$

2. We use the second (limit) comparison test. We have

$$
\frac{n^{3}}{\left(n^{2}+\sqrt{n}+1\right)^{x+1}} \sim \frac{1}{n^{2(x+1)-3}} \quad \text { as } n \rightarrow \infty
$$

Since for $2(x+1)-3>1$ the series

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

converges, and for $2(x+1)-3 \leq 1$ it diverges (as a generalized harmonic series), then by the second comparison test the series

$$
\sum_{n=1}^{\infty} \frac{n^{3}}{\left(n^{2}+\sqrt{n}+1\right)^{x+1}}
$$

converges for all $x>1$.

Answer. The interval of convergence of the series is $(1, \infty)$.