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 + [itex]\frac{1}{x}[/itex] + [itex]\frac{1}{x^2}[/itex] + [itex]\frac{1}{x^3}[/itex] + ...

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

Homework Equations

S[itex]\infty[/itex] = [itex]\frac{a}{1 - r}[/itex]

The Attempt at a Solution

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

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

Ans.: From textbook: -[itex]\frac{3}{2}[/itex] < x < [itex]\frac{3}{2}[/itex]

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.