QuestionJuly 14, 2025

Suppose that sum _(n=1)^infty a_(n)(x-8)^n converges at x=12 At which of the following values of z must the series also converge? 14 11 7 2 0 -2 -7 -11 -14

Suppose that sum _(n=1)^infty a_(n)(x-8)^n converges at x=12 At which of the following values of z must the series also converge? 14 11 7 2 0 -2 -7 -11 -14
Suppose that sum _(n=1)^infty a_(n)(x-8)^n converges at x=12 At which of the following values of z must the series also converge? 14 11 7 2 0 -2 -7 -11 -14

Solution
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Answer

11, 7, 2 Explanation 1. Determine the radius of convergence The series \sum _{n=1}^{\infty }a_{n}(x-8)^{n} converges at x = 12. Calculate the distance from the center to this point: |12 - 8| = 4. This is within the radius of convergence. 2. Identify interval of convergence The series converges for |x - 8| < 4. Therefore, the interval of convergence is (4, 12). 3. Check given values against interval Values that fall within (4, 12) are 11, 7, and 2.

Explanation

1. Determine the radius of convergence<br /> The series $\sum _{n=1}^{\infty }a_{n}(x-8)^{n}$ converges at $x = 12$. Calculate the distance from the center to this point: $|12 - 8| = 4$. This is within the radius of convergence.<br />2. Identify interval of convergence<br /> The series converges for $|x - 8| < 4$. Therefore, the interval of convergence is $(4, 12)$.<br />3. Check given values against interval<br /> Values that fall within $(4, 12)$ are $11$, $7$, and $2$.
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