- #1
odolwa99
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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.