Find a Formula for the Nth Partial Sum of the Series

question 62

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.
- $$\frac { 4 } { 2 \cdot 3 } + \frac { 4 } { 3 \cdot 4 } + \frac { 4 } { 4 \cdot 5 } + \ldots + \frac { 4 } { ( n + 1 ) ( n + 2 ) } + \ldots$$

• A) \(\frac { 4 \mathrm { n } } { \mathrm { n } + 2 } ; 4\)
• B) \(\frac { 4 \mathrm { n } } { 2 ( \mathrm { n } + 2 ) } ; 2\)
• C) \(\frac { 4 n } { 2 ( n + 1 ) } ; 2\)
• D) \(\frac { 4 n } { n + 1 } ; 4\)

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Definitions:

Objective Function

A mathematical expression that defines the goal of an optimization problem, e.g., to maximize profits or minimize costs.

Solved

A situation or problem that has been resolved, answered, or dealt with successfully.

Shadow Price

An estimated price for something that is not normally bought or sold in the market, often used in constrained optimization to value the cost of using an additional unit of a scarce resource.

Maximization Problem

A type of optimization that involves finding the maximum value of a function within a given domain, often subject to constraints.