I'm facing problems figuring out which values of $\alpha$ makes $$ \int_{1}^{\infty}\left(\sum_{j = 1}^{10}x^{j}\right) ^{-\left(n +1\right)/10}\,\frac{-n + 1}{10}\,x^{\alpha}\,\mathrm{d}x $$ convergent. The problem also states that $n>0$ and $\alpha$ is a real number.
I tried substituting the summation with the geometric series formula but I cannot figure out how to deal with the $n$.
The power of $x$ in each part of the integral must be less than $-1$ for the integral to converge. The powers are $$j\frac{-n+1}{10}+\alpha$$ for $j$ from $1$ to $10$
When $\frac{-n+1}{10}$ is positive, the power will be greatest at $j=10$, therefore $$-n+1+\alpha<-1$$$$\alpha<n-2$$
When $\frac{-n+1}{10}$ is negative, the power will be greatest at $j=1$, therefore $$\frac{-n+1}{10}+\alpha<-1$$$$\alpha<\frac{n-11}{10}$$