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# geometric series

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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

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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

May 2, 2021