Consider the Series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n } { n ^ { 2 } + 1 }$

question 187

Essay

Consider the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n } { n ^ { 2 } + 1 }$ .(a) Show that the series is convergent, but not absolutely convergent.(b) Calculate the sum of the first 8 terms to approximate the sum of the series.(c) Is the approximation in part (b) an overestimate or an underestimate?
(d) Estimate the error involved in the approximation from part (b).

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Definitions:

Withdrawal

The discomfort and distress that follow discontinuing an addictive drug or behavior.

Reinforcement

In behavioral psychology, a consequence applied to an organism's environment that increases the likelihood of that organism repeating the associated behavior in the future.

Shaping Behavior

A method of positive reinforcement of behaviors that are successively closer to the desired behavior, used in behavior modification programs.

Positive Reinforcers

Stimuli that, when introduced following a behavior, increase the likelihood of that behavior being repeated in the future.