# Thread: Sum of oscillating sequences

1. ## Sum of oscillating sequences

Find two oscillating sequences such that the sum of those two sequences diverges to infinity or to minus infinity, if possible. I have been unable to find two such sequences. It is easy enough to find two oscillating sequences who sum converges, but not diverges. Any ideas? Thanks.

2. take two sequences to be the same diverging oscillating sequence?

3. Originally Posted by zzzhhh
take two sequences to be the same diverging oscillating sequence?
But then you get an alternating sequence, which doesn't converge to infinity.

-1, 2, -1/3, 4, -1/5, 6, ...

and

1, -1/2, 3, -1/4, 5, -1/6, ...

4. By "oscillating" I assume that you mean alternating, i.e., the sign changes like $a_n > 0 \Rightarrow a_{n+1}<0$.

Think about what you want to do: a single alternating sequence can diverge, but it cannot have $\infty$ or $-\infty$ as a limit, because of the sign changes. The idea would be to find a sequence $(a_n)$ whose subsequence of even terms approach $\infty$ but its odd subsequence (of negative terms) remain bounded. Then find a sequence $(b_n)$ whose odd terms approach $\infty$ but its even terms (here, negative as per requirement) remain bounded. The sum $a_n + b_n$ should do what you want.

If this is too abstract, see Bruno J's suggestion.