# Sum to infinity

• January 24th 2008, 02:59 AM
SengNee
Sum to infinity
A geometric series has a common ratio, $r=\frac{b-a}{b}$ , b>0.
If 0<a<2b, show that the geometric series has a sum to infinity.
• January 24th 2008, 03:17 AM
mr fantastic
Quote:

Originally Posted by SengNee
A geometric series has a common ratio, $r=\frac{b-a}{b}$ , b>0.
If 0<a<2b, show that the geometric series has a sum to infinity.

You need to show that -1 < r < 1:

$0 < a < 2b \Rightarrow \frac{b - 2b}{b} < \frac{b - a}{b} < \frac{b - 0}{b} \equiv -1 < \frac{b - a}{b} < 1 \equiv -1 < r < 1$.