Answer
So, $b_m$ diverges.
Work Step by Step
We have
$$
\lim _{m \rightarrow \infty} b_m=\lim _{m \rightarrow \infty} 1+(-1)^m.
$$
It is clear that if $m$ is even then $\lim _{m \rightarrow \infty} b_m=2$ and if if $m$ is odd then $\lim _{m \rightarrow \infty} b_m=0$. Hence, the limit does not exist. So, $b_m$ diverges.
