Find a Formula for the Nth Partial Sum of the Series

question 257

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.
- $$\frac { 18 } { 1 ^ { 2 } \cdot 2 ^ { 2 } } + \frac { 30 } { 2 ^ { 2 } \cdot 3 ^ { 2 } } + \frac { 42 } { 3 ^ { 2 } \cdot 4 ^ { 2 } } + \ldots + \frac { 6 ( 2 n + 1 ) } { n ^ { 2 } ( n + 1 ) ^ { 2 } } + \ldots$$

• A) \(\frac { 6 n ^ { 2 } } { ( n + 1 ) ( n + 2 ) } ; 6\)
• B) \(\frac { 6 n ( n + 1 ) } { ( n + 2 ) ^ { 2 } } ; 6\)
• C) \(\frac { 6 ( \mathrm { n } + 1 ) ( \mathrm { n } + 2 ) } { \mathrm { n } ^ { 2 } } ; 6\)
• D) \(\frac { 6 n ( n + 2 ) } { ( n + 1 ) ^ { 2 } } ; 6\)

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Definitions:

Labor Supply Curve

A graphical representation that shows the relationship between the quantity of labor willing to be provided and the wage rate.

Nonlabor Income

Income received from sources other than direct employment or work, such as investments, pensions, or transfer payments.

Substitution Effect

The change in the quantity demanded of a good as consumers switch away from more expensive alternatives.

Income Effect

The change in consumer demand for goods and services induced by a change in consumers' discretionary income.