Consider the geometric series [ 5+10 x+20 x^{2}+40 x^{3}+cdots ] (a) Find the values of ( x ) for which this series converges. The limiting sum of this series is 100 . (b) Find the value of ( x ).
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The common ratio is found by dividing any term by the previous term. In this case, we have: \[ r = \frac{{10x}}{{5}} = \frac{{20x^2}}{{10x}} = \frac{{40x^3}}{{20x^2}} = \frac{{2x}}{{1}} = 2x \] Since the common ratio \( r \) must be between -1 and 1 for the Show more…
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