Limit of a sequence (2)

Calculus Level 5

Find the value of $\displaystyle\prod_{k=1}^{\infty} \left(1+\dfrac{1}{t_k}\right)$

Where $t_1=1$ and $t_n=n (1+t_{n-1})$

Also try limit as a sequence (1) and (3)

$\textbf{Assumptions and Source}$

$n \in \mathbb{N}-\{1\}$
This too is a standard problem in most calculus textbooks for $\text{JEE (Advanced)}$ .