Answer
Diverges
Work Step by Step
$$\lim _{n\rightarrow \infty }\left( 1+\dfrac {k}{n}\right) ^{n}=1^{n}+n\times 1^{n-1}\left( \dfrac {k}{n}\right) ^{1}+\ldots =1+k\neq 0$$
$$\Rightarrow \sum ^{\infty }_{n=1}\left( 1+\dfrac {k}{n}\right) ^{n}$$ diverges