Sum to Infinity of a Geometric Series

[odolwa99]

Homework Statement

Q. Find the range of values of x for which the sum to infinity exists for each of these series:

(i) 1 + \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + ...

(ii) \frac{1}{3} + \frac{2x}{9} + \frac{4x^2}{27} + \frac{8x^3}{81} + ...

Homework Equations

S\infty = \frac{a}{1 - r}

The Attempt at a Solution

(i) r = \frac{1}{x}/ 1 = \frac{1}{x} \Rightarrow 1 = x
Ans.: From textbook: IxI > 1

(ii) r = \frac{2x}{9}/ \frac{1}{3} = \frac{6x}{9} \Rightarrow 6x = 9 \Rightarrow x = \frac{9}{6} \Rightarrow x = \frac{3}{2}

Ans.: From textbook: -\frac{3}{2} < x < \frac{3}{2}

I'm confused as to whether I'm approaching this correctly, or if I've simply gone wrong in expressing the answers I found. Can someone help me figure this out? Thanks.

 

[rock.freak667]

For your sum to infinity to exist

S_n must converge as r→∞.

i.e. for |r| < 1

so in your first one, you correctly found r as r = 1/x so it would converge for |1/x| < 1 and you know that |X|< A ⇒ -A<X<A.

 

[odolwa99]

Ok, I think I see it now. Thanks for clearing that up.