Give an example of two series Pan and Pbn, both convergent,such that Panbn diverges

Answer 1

The key for this problem is to realize what happens when you multiply two alternating series together. For an alternating series to converge it's limit needs to be 0. This is significantly easier to achieve than for a nonalternating series. When you multiply two alternating series together, you get either a strictly negative function or a strictly positive function. So now the question is a lot easier, find a series that is convergent when alternating and divergent when not alternating. We will also try to find a convergent series that we can square to get the divergent series - that way we only need one series and can let an=bn

An easy solution that comes to mind is 1/n. ((-1)^n)/n is convergent while 1/n is divergent. To get a final solution we just need to adjust ((-1)^n)/n such that its square will equal 1/n while keeping adjusted form convergent. Since we are looking for a square, we can just take the square root to get ((-1)^n)/sqrt(n).