Conditionally Convergent Series?

Question

Well, I already know that Riemann proved that a conditionally convergent infinite series can be rearranged to converge to any real number as well as diverge to either positive or minus infinity. But specifically how is this done in an example? For example if I want the alternating harmonic (which "usually" converges to ln(2)) to converge to 5, how can I rearrange the terms?

In addition, does anyone know of any conditionally convergent series that are NOT alternating. It seems to me from the definition that it doesn't always have to be alternating to be conditionally convergent.

Answer 1

(i) Take positive elements of the alternating conditionally convergent, but NOT absolutely conv. series (the harmonic one, say) until the sum of those elements is more than 5. this can be done otherwise the subseries of positive terms would be bounded and thus the series would be absolutely convergent;

(ii) Now take negative elements and substract from the above until you get less than 5. Again, this is possible since otherwise the series would abs. conv.
Continue as above. Pay attention that at each step you pass over and then pass under 5 by less and less difference, since the terms you add (substract) are, in absolute value, lower and lower (since the general sequence of the series converges to zero!).
The above is just the way that, in general, Riemann's Theorem is proved.

The series cos(n)/n is conditionally convergent, but it isn't alternating.

Regards
Tonio