Provide an Appropriate Response $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }$

question 465

Multiple Choice

Provide an appropriate response.
-Find the sum of the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }$ by expressing $\frac { 1 } { 1 + x }$ as a geometric series, differentiating both sides of the resulting equation with respect to $x$ , multiplying both sides by $x$ , differentiating again, and replacing $\times$ by $\frac { 1 } { 3 }$ .

• A) \(\frac { 9 } { 32 }\)
• B) \(\frac { 9 } { 8 }\)
• C) \(\frac { 3 } { 32 }\)
• D) \(\frac { 3 } { 16 }\)

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Definitions:

Contact Hypothesis

The idea that regular interaction between members of different groups reduces prejudice, providing that it occurs under favorable conditions.

Prejudice

A preconceived opinion that is not based on reason or actual experience; often directed towards people or groups.

Stereotyping

The act of assigning fixed, generalized attributes to a group of people, ignoring individual differences.

Mental Energy

The capacity to apply cognitive resources towards thinking, problem-solving, and decision-making.