3. Let the general term formula of the sequence $\left\{a_{n}\right\}$ be $a_{n}=n^{2}+$ $\lambda n(n \in \mathbb{N})$. If $\left\{a_{n}\right\}$ is a monotonically increasing sequence, then the range of the real number $\lambda$ is
$3 \cdot \lambda > -3$.
$\because$ the sequence $\left\{a_{n}\right\}$ is a monotonically increasing sequence,
$\therefore a_{n+1} - a_{n} >$ C $n \in \mathbb{N}$; always holds.
Also, $a_{n} = n^{2} + \lambda n (n \in \mathbb{N})$.
$\therefore (n+1)^{2} + \lambda(n+1) - \left(n^{2} + \lambda n\right) > 0$ always holds,
which means $2n + 1 + \lambda > 0$.
Thus, $\lambda > -(2n + 1) (n \in \mathbb{N})$ always holds.
Since $n \in \mathbb{N}, -(2n + 1)_{\max} = -3 (n=1)$, therefore, $\lambda > -3$ is the required condition.