Sum to infinity

**SengNee** · January 24th 2008, 02:59 AM

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.

**mr fantastic** · January 24th 2008, 03:17 AM

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:

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

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