Infinite Series (2 diverge -> 1 converge)

[linuxux]

Infinite Series (2 diverge --> 1 converge)

I've been trying to figure this question out:

Find examples of two positive and decreasing series, \sum a_n and \sum b_n, both of which diverge, but for which \sum min(a_n,b_n) converges.

It doesn't make any sense to me that any positive and decreasing divergent series can be combined with another to produce a convergent series. Thanks in advance.

 

[Zurtex]

Are you sure you don't mean 2 positive and decreasing sequences a_n and b_n such that \sum a_n and \sum b_n diverge?

Just asking before I give it a crack.

 

[linuxux]

Are you sure you don't mean 2 positive and decreasing sequences a_n and b_n such that \sum a_n and \sum b_n diverge?

Just asking before I give it a crack.

100% sure.

 

[Zurtex]

Off the top of my head I'd say the question is flawed. If:

\sum_{n=0}^{i} a_n

Is a positive, decreasing divergent series in i, then WLOG we can say that a₀ >> 0. We can also say that a_n<0 for all n > 0. So if we take b_n = -a_n and look at this sequence:

\sum_{n=1}^{i} b_n

All summation terms are positive, the series doesn't converge and the series is strictly less than infinity. Now if the series is strictly less than infinity it is necessary that:

\lim_{n \rightarrow \infty} b_n = 0

And that for some n > N we have that:

\sum_{n=N}^{\infty} b_n &lt; B_0 \quad B_0 \in \mathbb{R}

But because b_n > 0 and the real numbers are complete, there must in fact be some B such that:

\sum_{n=N}^{\infty} b_n = B \quad B \in \mathbb{R}

(All the rigorous proof words escape my mind right now, but I'm quite confident this holds, it's reminding me of some work I did on my metric spaces course).

 

[linuxux]

thats not the first time I've come across a flawed question like this.
...thanks anyway.

 

[morphism]

thats not the first time I've come across a flawed question like this.
...thanks anyway.

What is a positive decreasing series?

I'm pretty certain that you mean a_n and b_n are decreasing sequences. (In which case the result is true.)

 

[linuxux]

What is a positive decreasing series?

I'm pretty certain that you mean a_n and b_n are decreasing sequences. (In which case the result is true.)

thats what I'm thinking. i am actually using the first version of a book that is now in its 8th revision, so I'm guessing there are a few errors. however i saw similar question in another book which did specify a_n and b_n being positive & decreasing sequences, while their series were divergent. I also need some advice on that problem.

Thanks.