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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.

 May 1, 2021
 #1
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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 smiley

 May 2, 2021

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