Show That the Series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { 3 n + 1 }$

question 333

Essay

Show that the series $\sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n - 1 } } { 3 n + 1 }$ is convergent. How many terms of the series do we need to add to find the sum to within 0.01?

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Definitions:

Self-instruction Training

A cognitive behavioral technique where one teaches themselves to direct attention, control behavior, and improve task performance through verbal cues.

Coping Self-statement

Positive affirmations or thoughts used by individuals to cope with stress, challenges, or to soothe themselves during difficult situations.

Managing Self-statement

The practice of consciously controlling one's internal dialogue or thought processes to influence behavior and emotions positively.

Factitious Disorder

A mental disorder in which an individual deliberately produces, feigns, or exaggerates symptoms of illness in themselves or others for any form of internal rewards, such as attention or sympathy.