Constructing a pair of sequences such that

[quarkycharmed]

Homework Statement

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}.

Homework Equations

none

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!

 

[mighty2000]

Thank you for your post. It seems like you are on the right track with your approach of using variants of 1/n. One way to construct the sequences {a_n} and {b_n} is to start with a_n = 1/n and b_n = 1/n^2. Both of these sequences are decreasing and divergent, but when you take the minimum of each term, c_n = min{a_n, b_n} = 1/n, which is a convergent series.

Another approach you can take is to use the alternating series test. For example, you can let a_n = (-1)^n/n and b_n = (-1)^n/n^2. Both of these sequences are decreasing and divergent, but when you take the minimum, c_n = min{a_n, b_n} = (-1)^n/n^2, which is a convergent series.

I hope this helps! Keep exploring different sequences and you will eventually find the solution. Remember to always check your work and make sure the conditions given in the problem are satisfied.
Scientist