# Sum of oscillating sequences

• Jun 7th 2010, 01:43 PM
tarheelborn
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.
• Jun 7th 2010, 05:38 PM
zzzhhh
take two sequences to be the same diverging oscillating sequence?
• Jun 7th 2010, 05:56 PM
Bruno J.
Quote:

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.

By "oscillating" I assume that you mean alternating, i.e., the sign changes like $\displaystyle a_n > 0 \Rightarrow a_{n+1}<0$.
Think about what you want to do: a single alternating sequence can diverge, but it cannot have $\displaystyle \infty$ or $\displaystyle -\infty$ as a limit, because of the sign changes. The idea would be to find a sequence $\displaystyle (a_n)$ whose subsequence of even terms approach $\displaystyle \infty$ but its odd subsequence (of negative terms) remain bounded. Then find a sequence $\displaystyle (b_n)$ whose odd terms approach $\displaystyle \infty$ but its even terms (here, negative as per requirement) remain bounded. The sum $\displaystyle a_n + b_n$ should do what you want.