Answer
$b_n$ converges to $1$.
Work Step by Step
We have
$$
\lim _{n \rightarrow \infty} b_n=\lim _{n\rightarrow \infty}
n(\ln(n+1)-\ln n)
\\
=\lim _{n\rightarrow \infty}
n\ln\frac{n+1}{n}=\lim _{n\rightarrow \infty}
n\ln\frac{n+1}{n}\\
=\lim _{n\rightarrow \infty}
\ln\left(1+\frac{1}{n}\right)^n=
\ln\lim _{n\rightarrow \infty} \left(1+\frac{1}{n}\right)^n=\ln e=1
$$
Where we used the fact that $\lim _{m \rightarrow \infty}
\left(1+\frac{1}{m}\right)^{m}=e$.
So, $b_n$ converges to $1$.
