Final answer:
The series ∞ (-2)ⁿ * xⁿ converges for values of x within the interval (-1, 1), which is determined by using the ratio test for convergence.
Step-by-step explanation:
To find the values of x for which the series ∞ (-2)ⁿ * xⁿ converges, we must consider the ratio test for convergence of an infinite series. The ratio test says that for the series ∞ aⁿ, it will converge absolutely if the limit as n approaches infinity of |aⁿ+1 / aⁿ| is less than 1. So, we apply the ratio test to our series (-2)ⁿ * xⁿ.
Let's calculate the limit:
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- |((-2)ⁿ+1 * xⁿ+1) / ((-2)ⁿ * xⁿ)|
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- = |(-2 * x) / (-2)|
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- = |x|
For the series to converge, we need |x| < 1. So, the interval of convergence for x is (-1, 1).