Answer
Suppose $\left\{ {{a_n}} \right\}$ converges.
Then we assume $\left\{ {{a_n} + {b_n}} \right\}$ converges. However, this contradicts $\left\{ {{b_n}} \right\}$ diverges. Hence, $\left\{ {{a_n} + {b_n}} \right\}$ diverges.
Work Step by Step
Suppose $\left\{ {{a_n}} \right\}$ converges to $L$, then by the limit definition:
$\mathop {\lim }\limits_{n \to \infty } {a_n} = L$
if, for every $\varepsilon > 0$, there is a number $M$ such that $\left| {{a_n} - L} \right| < \varepsilon $ for all $n>M$.
Now, we assume that $\left\{ {{a_n} + {b_n}} \right\}$ converges to $P$, then by the limit definition:
$\mathop {\lim }\limits_{n \to \infty } \left( {{a_n} + {b_n}} \right) = P$
if, for every $\delta > 0$, there is a number $N$ such that $\left| {{a_n} + {b_n} - P} \right| < \delta $ for all $n>N$.
Next, we write
$P = \mathop {\lim }\limits_{n \to \infty } \left( {{a_n} + {b_n}} \right) = \mathop {\lim }\limits_{n \to \infty } {a_n} + \mathop {\lim }\limits_{n \to \infty } {b_n}$
Since $\mathop {\lim }\limits_{n \to \infty } {a_n} = L$, we have
$\mathop {\lim }\limits_{n \to \infty } {b_n} = P - L$
This implies that the sequence $\left\{ {{b_n}} \right\}$ converges to a limit $P-L$. However, this contradicts the given statement that $\left\{ {{b_n}} \right\}$ diverges. Hence, $\left\{ {{a_n} + {b_n}} \right\}$ must diverge.
