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:

$$\sum_{n=N}^{\infty} b_n < B_0 \quad B_0 \in \mathbb{R}$$

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.