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The infinite geometric series x^2 - 2x^4 + 4x^6 - ... is equal to 1. Find the sum of all possible real values of x.

 Aug 11, 2023
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An infinite geometric series has a defined sum \(S = \frac{a_1}{1 - r}\) where a_1 is the first term and r is the common ratio as long as |r| < 1.

To obtain the next term in the series \(x^2 - 2x^4 + 4x^6 - \cdots \), multiply the previous term by -2x^2, which is the common ratio. We know that the series equals 1, so we can put this information together to find possible values of x.

 

\(S = \frac{a_1}{1 - r} \\ 1 = \frac{x^2}{1 - (-2x^2)} \\ 2x^2 + 1 = x^2 \\ x^2 + 1 = 0 \\ x^2 = -1\)

 

At this point, we need not go further as there are no real-valued solutions to x^2 = -1. No real values of x make the series equal 1. I suppose this means that the sum of all values of x is 0?

 Aug 12, 2023
edited by The3Mathketeers  Aug 12, 2023

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