When this series converges? $\sum_{n=45}^{\infty}(-1)^n(x^2+2x)^{\log({R(n)})}$ with R(n)=... For n=1,2,.. is $P(n)=n^{10}-\sum_{k=1}^9{\ k  n^k}$ and $Q(n)=\sum_{k=1}^8{\   n^k}$ and is $R(n)=\dfrac{P(n)}{Q(n)}$. 
Determine for which values ​​of $x$ the series

$$\sum_{n=45}^{\infty}(-1)^n(x^2+2x)^{\log({R(n)})}$$

is convergent,  distinguishing simple and absolute convergence.

My answer:
Is $a=x^2+2x$. I wrote that for $|a|&lt;1$ converges absolutely. Necessary condition for convergence is also simple that the limit of $(x^2+2x)^{\log({R(n)})}$ is $0$: being that if |a|>1 the limit is different from $0$ get that converges only for $|a|&lt;1$. Is that okay?
Hint: For $x^2 + 2 x &gt; 0$, you'll want to think about alternating series. For $x^2 + 2 x &lt; 0$, things are rather more complicated because $(x^2+2x)^{\log(R(n))}$ is complex, but absolute convergence should be easy to test because $\log(R(n)) = 2 \log(n) + o(1)$.
Hint: For $x^2 + 2 x &gt; 0$, you'll want to think about alternating series. For $x^2 + 2 x &lt; 0$, things are rather more complicated because $(x^2+2x)^{\log(R(n))}$ is complex, but absolute convergence should be easy to test because $\log(R(n)) = 2 \log(n) + o(1)$.