True, the series is indeed divergent.
The <em>n</em>-th term is given by
(<em>n</em> + 1) / (<em>n</em> + 2)
starting with <em>n</em> = 1, so that
(<em>n</em> + 1) / (<em>n</em> + 2) = (<em>n</em> + 2) / (<em>n</em> + 2) - 1 / (<em>n</em> + 2) = 1 - 1 / (<em>n</em> + 2)
and as <em>n</em> grows to infinity, the rational term vanishes, the summand converges to 1, and the sum behaves like ∑ 1, which diverges.