Find the Power Series for the Function $f ( x ) = \frac { 8 x } { x + 1 }$

question 113

Multiple Choice

Find the power series for the function $f ( x ) = \frac { 8 x } { x + 1 }$ using the power series for $\frac { 1 } { x + 1 }$ .

• A) \(\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n + 1 } 8 x ^ { n + 1 }\)
• B) \(\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n - 1 }\)
• C) \(\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } 8 x ^ { n - 1 }\)
• D) \(\sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n + 1 }\)
• E) \(\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n } 8 x ^ { n + 1 }\)

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Definitions:

Reactance Theory

A psychological theory suggesting that people act against threats to their freedoms or choices to maintain a sense of control.

Psychodynamic Theory

A psychological perspective that emphasizes unconscious processes and childhood experiences in shaping behavior and personality.

Systems Theory

An interdisciplinary theory that analyzes systems—a set of interacting or interdependent components—as a whole to understand problems, make predictions, and guide improvements.

Cognitive Behavioral Therapy

Cognitive Behavioral Therapy (CBT) is a psychotherapeutic intervention that aims to improve mental health by challenging and changing unhelpful cognitive distortions and behaviors, improving emotional regulation, and developing personal coping strategies.