Answer
Consider:
${a_n} = \ln {2^n}$ and ${b_n} = \ln {2^n} - 2$.
${a_n}$ and ${b_n}$ diverge. However, $\left\{ {{a_n} + {b_n}} \right\}$ converges to $-2$.
Work Step by Step
Consider the sequences: ${a_n} = \ln {2^n}$ and ${b_n} = \ln {2^n} - 2$.
Evaluate the limit of ${a_n}$:
$\mathop {\lim }\limits_{n \to \infty } \ln {2^n} = \mathop {\lim }\limits_{n \to \infty } n\cdot\ln 2 = \left( {\ln 2} \right)\cdot\mathop {\lim }\limits_{n \to \infty } n = \infty $.
Thus, ${a_n}$ diverges.
Evaluate the limit of ${b_n}$:
$\mathop {\lim }\limits_{n \to \infty } \left( {\ln {2^n} - 2} \right) = \mathop {\lim }\limits_{n \to \infty } n\cdot\ln 2 - \mathop {\lim }\limits_{n \to \infty } 2 = \left( {\ln 2} \right)\cdot\left( {\mathop {\lim }\limits_{n \to \infty } n} \right) - 2 = \infty $.
Thus, ${b_n}$ diverges.
However, ${a_n} + {b_n} = - 2$. All of the terms in the sequence $\left\{ {{a_n} + {b_n}} \right\}$ are equal to $-2$. So, it converges to $-2$.
