Converging Geometric Series with Negative Values?

[Lancelot1]

Hiya everyone,

Alright ?

I have a simple theoretical question. In a decreasing geometric series, is it true to say that the ratio q has to be 0<q<1, assuming that all members of the series are positive ? What if they weren't all positive ?

Thank you in advance !

 

[HallsofIvy]

A "geometric series" is of the form $$\sum ar^n= a+ ar+ ar^2+ ar^3+ \cdot\cdot\cdot= a(1+ r+ r^2+ r^3+ \cdot\cdot\cdot)$$ so, yes, if the series is decreasing and positive then r must be less than 1. If r> 1 then $$1< r< r^2< r^3< \cdot\cdot\cdot$$. If r< 1 then $$1> r> r^2> r^3> \cdot\cdot\cdot$$. Of course, in the first case, r> 1, the series does not converge.

If they are not all positive then either they are all negative and we can take the negative into the "a" term so that $$\sum at^n= a\sum r^n$$ has a negative number times the same sum of $$ar^n$$ or they are alternating, $$\sum a(-r)^n= a\sum (-1)^n r^n$$. The geometric series $$\sum ar^n$$ converges if and only if $$-1< r< 1$$.