This is only possible for aconditionallyconvergent series- that is, a series having both positive and negative terms such that converges but doesnotconverge.

We start by separating positive and negative "parts". Specifically, we define if , if and then define if , if . Note that both and are positive sequences.

It is then easy to see that and . (There isn't really a sum there- for all n one of the terms in each of those is 0.)

Now, what about the convergence of and ? If both series converged, then, since , we would have converges which is not true so the cannotbothconverge.

Suppose one converged and the other didn't? If converges we have that so converges- but we already know they cannot both converge. Similarly, if we assume converges, we can show that also converges, which cannot happen.

That is, both and must diverge and, since they are both series of positive numbers must diverge to positive infinity- and that means they are unbounded.

Given any integer M, there is some such that . We can also find such that . The sum of those two sums is between M/2- 1 and M/2. But if we remove a finite number of terms from a divergent series, the terms left still diverge. We can do this again. And after that, still again, finding a finite number of terms from and , that is, terms from that always converge to a number between M/2-1 and M/2. But that means that we can order all of the numbers in so that the sum becomes larger and larger- it diverges.

In fact, doing basically that same process, we can rearrange the terms in the sequence to converge to any number at all!