Is ∑ₖ₌₀ ∞ (5x − 20)ᵏ a power series? If so, find the center a of the power series and state a formula for the coefficients cₖ of the power series.

검증된 단계별 안내

Recognize that a power series is generally expressed in the form \(\sum_{k=0}^\infty c_k (x - a)^k\), where \(a\) is the center of the series and \(c_k\) are the coefficients depending only on \(k\).

Rewrite the given series \(\sum_{k=0}^\infty (5x - 20)^k\) by factoring the expression inside the power to isolate \(x\): note that \(5x - 20 = 5(x - 4)\).

Express the series as \(\sum_{k=0}^\infty [5(x - 4)]^k = \sum_{k=0}^\infty 5^k (x - 4)^k\) to match the standard power series form.

Identify the center \(a\) of the power series as the value of \(x\) that makes the term inside the parentheses zero, which is \(a = 4\).

From the rewritten series, determine the coefficients \(c_k\) as \(c_k = 5^k\), since the series is now \(\sum_{k=0}^\infty c_k (x - 4)^k\).

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주요 개념

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Power Series Definition

A power series is an infinite series of the form ∑ₖ₌₀ ∞ cₖ(x - a)ᵏ, where cₖ are coefficients and a is the center. It represents a function as a sum of powers of (x - a), converging within a certain radius. Recognizing the structure helps determine if a given series qualifies as a power series.

Center of a Power Series

The center 'a' of a power series is the value around which the series is expanded, appearing in the term (x - a). Identifying 'a' involves rewriting the series so that the variable part matches (x - a)ᵏ. This center is crucial for understanding the interval of convergence and the behavior of the series.

Coefficients of a Power Series

The coefficients cₖ in a power series determine the weight of each term (x - a)ᵏ. Finding cₖ often requires expressing the series in the standard form and comparing terms. These coefficients are essential for analyzing the function represented and for operations like differentiation or integration of the series.