Question

Let {an} ∞ n=1 and {bn} ∞ n=1 be sequences both of which diverge to −∞. Then {an + bn} ∞ n=1 diverge to −∞ and {an · bn} ∞ n=1 diverge to +∞.

Answer 1

Solution:

Statement: If $\{a_n\}\ and\ \{b_n\}$ are divergent then $\{a_n + b_n\}$ is divergent.

It is false.

Example: $\{a_n\} = n, \{b_n\} = −n, \{a_n + b_n\} = 0$ , which is convergent.

Statement: If $\{a_n\}\ and\ \{b_n\}$ are divergent then $\{a_n b_n\}$ is divergent. .

It is false.

Example: $\{a_n\} = \{b_n\} = (−1)^n, \{a_n b_n\} = 1$ , which is convergent.