Finding the convergence range for 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.