# For what values of x will the infinite geometric series 1+ (2x-1) + (2x-1)^2 + (2x-1)^3 + ... have a finite sum?

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The infinite geometric series will have a finite sum if the common ratio ( |r| < 1 ). Therefore, for the series ( 1 + (2x - 1) + (2x - 1)^2 + (2x - 1)^3 + \ldots ) to have a finite sum, the absolute value of the common ratio ( |2x - 1| ) must be less than 1. This gives us the inequality ( |2x - 1| < 1 ). Solving this inequality gives ( \frac{-1}{2} < x < \frac{3}{2} ). Therefore, the values of ( x ) for which the infinite geometric series has a finite sum are ( x ) such that ( \frac{-1}{2} < x < \frac{3}{2} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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