Use the Power Series $\frac { 1 } { 1 - x } = \sum _ { n = 0 } ^ { \infty } x ^ { n } , | x | < 1$

question 98

Multiple Choice

Use the power series $\frac { 1 } { 1 - x } = \sum _ { n = 0 } ^ { \infty } x ^ { n } , | x | < 1$
to determine a power series for the function $f ( x ) = \frac { 5 x } { ( 1 - 5 x ) ^ { 2 } }$ .

• A) \(\sum _ { n = 1 } ^ { \infty } n ( - 1 ) ^ { n } ( 5 x ) ^ { n + 1 }\)
• B) \(\sum _ { n = 1 } ^ { \infty } n ( - 1 ) ^ { n } ( 5 x ) ^ { n }\)
• C) \(\sum _ { n = 1 } ^ { \infty } n ( 5 x ) ^ { n }\)
• D) \(\sum _ { n = 1 } ^ { \infty } n ( 5 x ) ^ { n - 1 }\)
• E) \(\sum _ { n = 1 } ^ { \infty } n ( 5 x ) ^ { n + 1 }\)

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Definitions:

System II

Often refers to a mode of thinking that is slower, more deliberative, and more logical than fast, automatic, intuitive responses (System I thinking).

Quantifiable

Capable of being measured or expressed as a quantity, enabling numerical assessment or comparison.

System II

A term often used in psychology to describe the deliberative, analytical form of cognition that requires concentration and is slower than intuitive thinking.

Intuitive Thinking

A cognitive process that involves making judgments and solving problems based on implicit knowledge and feelings, rather than explicit reasoning.