MHB 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&lt; r&lt; r^2&lt; r^3&lt; \cdot\cdot\cdot. If r< 1 then 1&gt; r&gt; r^2&gt; r^3&gt; \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&lt; r&lt; 1.