Do the Following Sequences Converge or Diverge? If They Converge $\left\{\frac{5 n}{2 n+1}\right\}$

question 27

Essay

Do the following sequences converge or diverge? If they converge, state the limit.
(a) $\left\{\frac{5 n}{2 n+1}\right\}$
(b) $\left\{\frac{n^{3}+1}{2 n+1}\right\}$
(c) $\left\{\frac{3 n^{3}}{4 n^{5}+1}\right\}$

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Definitions:

Self-efficacy

The belief in one's abilities to succeed in specific situations or accomplish a task.

Behavior Change

The process of transforming one's actions and attitudes towards healthier habits, often key in improving personal well-being and health outcomes.

Overcoming Obstacles

The process or act of resolving or surpassing barriers or difficulties to achieve a goal.

Transtheoretical Model

A theory of behavior change that evaluates an individual's readiness to act on a new healthier behavior, and provides strategies or processes to guide the individual.