Answer
Diverges
Work Step by Step
We can write the general form of the given series as: $a_k=(\dfrac{10k^3+k}{9k^3+k+1})^k$
Root Test states that when $\Sigma a_k$ is an infinite series with positive terms and, then $r=\lim\limits_{k \to \infty}\sqrt[k] a_k$
a) When $0 \leq r \lt 1$, the series converges. (b) When $r \gt 1$, or, $\infty$, so the series diverges. (c) When $r=1$, the ratio test is inconclusive.
Now, $r=\lim\limits_{k \to \infty}\sqrt[k] {(\dfrac{10k^3+k}{9k^3+k+1})^k} \\= \lim\limits_{k \to \infty} \dfrac{10k^3+k}{9k^3+k+1}k \\=\dfrac{10}{9}$
Therefore, the series diverges by the root test.
