(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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] + ...

2. Relevant equations

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

3. 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 text book: 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 text book: -[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.

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# Sum to Infinity of a Geometric Series

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