Answer
Diverges
Work Step by Step
We can write the general form of the given series as: $a_k=(1+\dfrac{3}{k})^{k^2}$
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, $r=\lim\limits_{k \to \infty}\sqrt[k] {(1+\dfrac{3}{k})^{k^2}} \\= \lim\limits_{k \to \infty} [(1+\dfrac{1}{k/3})^{k/3}]^3 \\=e^3$
Therefore, the series diverges by the root test.
