> Randall-- > > Thank you for your repsonse to my probably pretty basic question on > the infinite series. > > Now if r is basically 0<r<1, and if we want the series to be an > incresing series, then we have to assume something slightly different, > correct? (I'm not sure I put this right but the series needs to start > with the smallest number larger than 0 and the progress toward 1 in > our series).
Yes, the series needs start with an initial term that is not zero. and 0< r < 1 (more on this later about r)
You bring up the point that we can have a negative term, let's use the example of the initial term a = -1 and r = 1/3
Then we have the series -1 -1/3 -1/9 -1/27 -1/81 and you can see that this is similar to a positive series, only it has a negative sum. You could factor the -1 from the series and get
and the partial sums are 2^n-1, which can be made as big as you like.
and for r <= -1, I am not sure if a sum even exists. Suppose we let a=1 and r = -2, then we get the alternating series:
1 -2 4 -8 16 -32 64 -128 256 -512 1024 -2048
This has the partial sums of 1,-1,3,-5,11,-21,43,-85,.... which are wildly diverging, so not much can be said other than the sums oscillate with increasing amplitude. That's why I've said that I wonder if an infinite sum even exists.
I hope this sheds some more light on series for you.