Find all the values of x such that the series \sum_{n=1}^\infty \frac{(5x-9)^n}{n^2} would...


Find all the values of {eq}x {/eq} such that the series {eq}\sum_{n=1}^\infty \frac{(5x-9)^n}{n^2} {/eq} would converge.

Interval of Convergence for a Power Series:

To find the interval of convergence {eq}( I ) {/eq}, we'll use the series ratio test. To check the endpoints, we'll use the definition of absolute/conditional convergence and the p-series test to check the endpoints, which states that:

If the series is of the form {eq}\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^{1/p}} {/eq}

If {eq}p>1 {/eq} then the p-series converges.

If {eq}0<p\leq1 {/eq} then the p-series diverges.

Answer and Explanation:

We are given: {eq}\displaystyle \sum_{n = 1}^{\infty} \frac{(5x-9)^n}{n^2} {/eq}

Use the Series Ratio test, which states that:

If there exists an N...

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from PSAT Prep: Tutoring Solution

Chapter 10 / Lesson 13

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