# Constructing a pair of sequences such that

1. Mar 23, 2012

### quarkycharmed

1. The problem statement, all variables and given/known data

The question asks me to construct a pair of sequences {a_n} from n=1 to inf and {b_n} from n=1 to inf such that a_n and b_n are both greater than or equal to zero for all integers n, both sequences are decreasing, and both series of these sequences diverge, but also such that the series c_n from n=1 to inf converges, where c_n is defined as the min{a_n,b_n}.

2. Relevant equations

none

3. The attempt at a solution

I've played around with a lot of sequences whose series I know to be convergent, mainly variants of 1/n, but I cannot figure out how to make two sequences such that when choosing the smaller nth term from the two sequences I wind up with a convergent series. The problem I see is that every convergent decreasing series I think of requires each term to get smaller much faster than any decreasing divergent series I can think of. I'm sure there is something really obvious I'm missing, so I don't know if anyone can give a hint without giving it away, but is there another angle from which I should look at this?

Thanks!