Find a Formula for the Nth Partial Sum of the Series

question 86

Multiple Choice

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
- $$\frac { 9 } { 1 ^ { 2 } \cdot 2 ^ { 2 } } + \frac { 15 } { 2 ^ { 2 } \cdot 3 ^ { 2 } } + \frac { 21 } { 3 ^ { 2 } \cdot 4 ^ { 2 } } + \ldots + \frac { 3 ( 2 n + 1 ) } { n ^ { 2 } ( n + 1 ) ^ { 2 } } + \ldots$$

• A) \(\frac { 3 n ( n + 2 ) } { ( n + 1 ) ^ { 2 } } ; 3\)
• B) \(\frac { 3 n ( n + 1 ) } { ( n + 2 ) ^ { 2 } } ; 3\)
• C) \(\frac { 3 n ^ { 2 } } { ( n + 1 ) ( n + 2 ) } ; 3\)
• D) \(\frac { 3 ( n + 1 ) ( n + 2 ) } { n ^ { 2 } } ; 3\)

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Definitions:

Positive Punishment

A method of behavior modification that involves the addition of an unfavorable outcome or stimulus following an undesirable behavior to decrease its occurrence.

Negative Punishment

A behavior modification technique where a favorable stimulus is removed after a certain behavior, decreasing the likelihood of that behavior occurring in the future.

Positive Reinforcement

A behavioral concept where a desirable stimulus is introduced following a behavior, increasing the likelihood that the behavior will occur again.

Negative Punishment

A behavior modification technique involving the removal of a desirable stimulus to decrease the likelihood of a behavior being repeated.