Proving the product of two series diverges to infinity, given that one series (An) converges to a limit L and (Bn) diverges to infinity, I have to prove that the product of the two series (AnBn) diverges to infinity.
I am not sure if the algebra of limits can be applied to diverging sequences, is it sufficient to say that lim(AnBn) = lim(An)lim(Bn) = infinityL, therefore the overall limit is also infinity?
Best Answer
Let $A_n = n$ and $B_n = 1/n$. Then the sequence $C_n = A_nB_n$ is constant, so certainly has a limit. But $A_n$ diverges to infinity and $B_n$ has a limit.
However, if $A_n$ diverges to infinity and $B_n$ has a nonzero limit, then certainly their product will diverge. To prove this, consider that if $B_n$ has a nonzero limit $B$, then beyond some $n$ all terms of $B_n$ will have magnitude greater than, say, $|B|/2$...