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Re: infinite series question
Posted:
Nov 24, 2002 12:18 PM


unrealistic wrote:
> 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
1 * ( 1 + 1/3 + 1/9 + 1/27 + 1/81 + ...) 1 * ( 3/2 ) 3/2
Can r be less than 0? Let's try it and see.
Something interesting happens when we pick 1 < r < 0, say 2/5. Then we have an alternating series. Let the first term be 1, and we then have
1 2/5 4/25 8/125 16/625 32/3125 + ..
Does this have a sum? If you work out this series long enough and add the terms, you will find that the sum is 5/7. (or use the formula)
> The way I understand it, what you suggested works for between 1 and 2 > (right?) but it we are looking at between 0 and 1, it has to be > different because of the issue with multiplying by 0.
Not necessarily. Suppose we pick a = 1/3 and r = 3/4
This is the series 1/3 1/4 3/16 9/64 27/256 ...
The sum is 4/3
So nothing is really special about a. We can pick a to be something quite large, say 1920, and r = 1/8
The series is 1920 240 30 15/4 15/32 15/256 15/2048 ... and the sum is 15360/7 or 2194 + 2/7
So nothing special about a, except that if we pick a = 0, then nothing happens. So the values to be avoided are a=0 and r=0.
> > Is there some special sort of notation I am missing here? >
Nope, in summary the initial term, a can be anything except 0, but 1 < r < 1 except we notice that at r=0, we don't have a series anymore, just the first term a.
When r >= 1 the series diverges, an example is a=1 and r=2, we get the sum of powers of 2
1 + 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + ...
and the partial sums are 2^n1, 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.
Randall



