 Suppose we have an absolutely convergent series. Let another series have the same terms but in a different order. Is our new series absolutely convergent? Well, suppose it isn't. Then there is some epsilon where for any n there is m greater than n greater than n where a particular sum is greater than epsilon. Now the argument after this point may get a little complicated so it may help to have some concrete, if made up, numbers. So suppose that our epsilon is 0.01 and for any n we can find m greater than n where the finite sum is greater than epsilon. Now since the original series is absolutely convergent then for this value of epsilon there is an n star where for all m greater than n greater than n star the sum is less than epsilon. So suppose again just for concreteness and so we can organize our thinking that n star is 5. So again the made up numbers are not part of our proof they are there to organize our thinking and in this case our ability to make up this number comes from the fact that our original series was assumed to be absolutely convergent. So in the proof we need to say something like this since our original series converges then for this epsilon there is some n star where for m greater than n greater than n star the finite sum will be less than epsilon. Since the terms of our new series have the same as those of the original series then the first n star equals 5 terms of our original series are somewhere in the terms of the new series. And since there's a finite number of them suppose that all of these terms appear at the first oh I don't know 100 or so terms of our new series which means that all of the remaining terms must correspond to terms in the original series with index greater than 5. So again these arbitrary numbers are not part of the proof the actual statement we'd want to say is that we want to find an n hat where if our index is greater than n hat then the term of the rearranged series corresponds to a term of the original series whose index is greater than n star. So remember we assumed our rearranged series was divergent so our assumption means that we can find an m and n greater than 100 where sum sum will be greater than epsilon. Again we'll throw down a concrete sum as a placeholder. Now by assumption these correspond to terms of our original series with index greater than 5 where we'll arrange these in order of increasing index and so epsilon is less than the sum but we can fill in the intervening values and this is a problem because remember we assumed our original series was absolutely convergent and this sum has to be less than epsilon as long as m and n are greater than 5 and they are. Again we made up these numbers so we could organize our thinking formally by assumption we have m and n greater than n hat where our sum of the rearranged series is greater than epsilon but since these terms correspond to terms of the original series with index greater than n star then we'll take the least and greatest index and so our sum of the rearranged series is less than a particular sum of the original series and that has to be false since our original series is absolutely convergent and this proves the important theorem if the series is absolutely convergent that any rearrangement of the terms is also going to be absolutely convergent and in fact we can prove that if the series is absolutely convergent and we have a rearrangement then the sum of the two series are the same but you should prove this.