Which of the series, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series’ convergence or divergence.) sum_{n=1}^inftyfrac{1}{2n-1}

Determine the radius of convergence and the interval of convergence for each power series.
$\sum _{n=0}^{\mathrm{\infty }}\sqrt{n}(x-1{)}^n$

Solving summation problem
${\displaystyle \sum _{i=x}^{2x}5-\sum _{k=2x+1}^{5x+2}5=-25}$

Find the interval of convergence of the power series.
$\sum _{n=0}^{\mathrm{\infty }}\frac{(3x{)}^n}{(2n)!}$

I am having difficulty finding if this series converges or diverges:
${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{{(-2)}^{n+1}}{n^{n-1}}}$
I am unsure of which test to use. At first, I thought I should use alternating series test, but I am unable to manipulate the numerator to ${\displaystyle {(-1)}^{n+1}}$. I thought I may be able to use root test, but I have no clue how to manipulate the series to do that.
What series test would I use, and how?

Find the sum of the series.
$\sum _{n=1}^{\mathrm{\infty }}\frac{2}{(n+8)(n+6)}$

Find the value of $\frac{1}{2(2^2-1)}+\frac{1}{3(3^2-1)}+\frac{1}{4(4^2-1)}+\cdots .$

Show that $\sum _{k=1}^{\mathrm{\infty }}{\sin }^k(k\pi /6)$ diverges?