Find all r for which the infinite geometric series 2 + r + r/2 + r^2/4 + ... is defined. Enter all possible values of r, as an interval.
+526 There's a slight correction in the given GP, shouldn't it be \(2+r+{r^2\over 2}+{r^3\over 4}+...?\)
As per the question,
First term of the GP \(a=2\)
and common ratio \(={r\over 2}\)
We know that sum of infinite GP
S∞ \(={a\over r-1}\)
⇒ S∞ \(={2\over {r\over 2}-1}\)
\(={4 \over r-2}\)
Now, S∞ is defined for \(-1 < r <1 \)
∴ All possible values of r are in the interval \((-1,1)\).
~Amy 
