Answer
The series diverges.
Work Step by Step
We have the $n$th partial sums: ${s_n} = \mathop \sum \limits_{k = 3}^n \dfrac{1}{{k - 2}}$. So,
${s_3} = 1$
${s_4} = 1 + \dfrac{1}{2}$
${s_5} = 1 + \dfrac{1}{2} + \dfrac{1}{3}$
${s_6} = 1 + \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{1}{4}$
...
So, the partial sums form a strictly increasing sequence such that
${s_3} \lt {s_4} \lt {s_5} \lt \cdot\cdot\cdot \lt {s_n} \lt \cdot\cdot\cdot$
Since there is no upper bound, by Theorem 9.2.3 part (b) in Section 9.2: $\mathop {\lim }\limits_{n \to \infty } {s_n} = \infty $. Therefore, the series diverges.
