SOLUTION: For which values of x will ((-3x+5)/2)+(((-3x+5)/2)^2)+(((-3x+5)/2)^3)+... Be a converging series?

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Question 1018776: For which values of x will ((-3x+5)/2)+(((-3x+5)/2)^2)+(((-3x+5)/2)^3)+... Be a converging series?
Found 2 solutions by ikleyn, rothauserc:
Answer by ikleyn(52790)   (Show Source): You can put this solution on YOUR website!
.
For which values of x will ((-3x+5)/2)+(((-3x+5)/2)^2)+(((-3x+5)/2)^3)+... Be a converging series?
----------------------------------------------------------- 

The necessary and sufficient condition is this inequality


 < 1,   or, which is the same, these two inequalities


-1 <  < 1.


Can you solve it?


Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
let u = ((-3x+5)/2), then we have
u + u^2 + u^3 + ... = (1 / (1 - u)) iff |u| < 1
:
| (-3x+5)/2) | < 1
:
: using the pattern for <, we clear the absolute value
:
1) -1 < (-3x+5)/2) < 1
-2 < -3x+5 < 2
-7 < -3x < -3
7/3 > x > 1
:
the interval is 1 < x < 7/3 on which ((-3x+5)/2) converges

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