1. The first term in a geometric series is x and the common ratio is x/(1+x). Find the set of values of x for which the series is convergent.
2.The first term in a geometric series is (x-9)/(x+5) and the common ratio is (x+5)/(x-9), where x =/= -5, x=/= 9. For what set of values of x are all the terms of the series positive? Find the set of values of x for which the series is convergent.
I really need some help in doing these, thanks very much!
You know that an (infinite) geometric series converges exactly when the common ratio has absolute value less than 1. So
You know that the common ratio must be positive for the whole thing to be positive, so
is necessary. Also, the first term must be positive, but it is just the reciprocal of the common ratio, so it is positive when the common ratio is, so you get:
For the second part, again you need the common ratio to be less than 1, so do the same procedure as I did for the first one, clearing the denominator, squaring, and solving.