Determine If the Series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n - 1 } { 4 n + 1 } \right) ^ { n }$

question 160

Essay

Determine if the series $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 n - 1 } { 4 n + 1 } \right) ^ { n }$ converges or diverges by the Ratio Test or Root Test.

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Definitions:

Cognitive Dissonance

The mental discomfort experienced by an individual who holds two or more contradictory beliefs, ideas, or values simultaneously.

Subjective Norm

The perceived social pressure to engage or not engage in a particular behavior, based on the individual's belief about what important others think they should do.

Belief Hypocrisy

The condition of holding beliefs that one's own behavior does not conform to, often leading to cognitive dissonance when recognized.

Cognitions

The mental processes involved in gaining knowledge and comprehension, including thinking, knowing, remembering, judging, and problem-solving.