The series $\displaystyle \sum_{n=1}^\infty n^k r^n$ converges when $0 < r < 1$ and diverges when $r>1$. This is true regardless of the value of the constant $k$. When $r =1$ the series is a p-series. It converges if $k < -1$ and diverges otherwise. Each of the series below can be compared to a series of the form $\displaystyle \sum_{n=1}^\infty n^k r^n$. For each series determine the best value of r and decide whether the series converges.
A. $\displaystyle \sum_{n=1}^\infty \ (3 +n( 5)^ n)^{- 3}$
$r$ = converges or diverges (c or d)?

B. $\displaystyle \sum_{n=1}^\infty n^{\pi} \frac{3^{2n}}{6 ^n + n^9}$
$r$ = converges or diverges (c or d)?

C. $\displaystyle \sum_{n=1}^\infty \frac{n^{.5} + 7 }{ n^{6} + 5 }$
$r$ = converges or diverges (c or d)?

D. $\displaystyle \sum_{n=1}^\infty \left( \frac{7 n^2 + 3 n + 3 ^{-3 n} }{6^ {n + 6 } + 6 n + 5 \sqrt n } \right) ^{3 }$
$r$ = converges or diverges (c or d)?