Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.
e. Diverges for x = 8

검증된 단계별 안내

Recall that the radius of convergence \( R \) of a power series \( \sum a_n x^n \) is the distance from the center (here assumed to be 0) within which the series converges absolutely.

Given that the series converges at \( x = 4 \), this means \( |4| = 4 \) is within or on the boundary of the radius of convergence, so \( R \geq 4 \).

Given that the series diverges at \( x = 7 \), this means \( |7| = 7 \) is outside the radius of convergence, so \( R < 7 \).

Since \( R \) is between 4 and 7, the behavior of the series at \( x = 8 \) (where \( |8| = 8 \)) depends on whether 8 is inside or outside the radius of convergence.

Because \( 8 > R \), the series must diverge at \( x = 8 \). Therefore, the statement 'Diverges for \( x = 8 \)' is true.

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주요 개념

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Radius of Convergence

The radius of convergence of a power series ∑ aₙxⁿ is the distance from the center (usually zero) within which the series converges absolutely. If the series converges at x = 4 but diverges at x = 7, the radius of convergence R satisfies 4 ≤ R < 7.

Behavior of Power Series Outside the Radius of Convergence

A power series diverges for all values of x whose absolute value exceeds the radius of convergence. Therefore, if |x| > R, the series must diverge. This helps determine convergence or divergence at points beyond the radius.

Insufficient Information on Boundary Points

Convergence at points where |x| equals the radius of convergence is not guaranteed and must be checked separately. Since divergence at x = 7 implies R < 7, the behavior at x = 8 depends on whether 8 lies inside or outside the radius, which can be inferred but not always conclusively.

Assume that the series ∑ aₙxⁿ converges for x = 4 and diverges for x = 7. Answer true (T), false (F), or not enough information given (N) for the following statements about the series.e. Diverges for x = 8