Provide an Appropriate Response $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }$

question 465

Multiple Choice

Provide an appropriate response.
-Find the sum of the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n ^ { 2 } } { 3 ^ { n - 1 } }$ by expressing $\frac { 1 } { 1 + x }$ as a geometric series, differentiating both sides of the resulting equation with respect to $x$ , multiplying both sides by $x$ , differentiating again, and replacing $\times$ by $\frac { 1 } { 3 }$ .

• A) \(\frac { 9 } { 32 }\)
• B) \(\frac { 9 } { 8 }\)
• C) \(\frac { 3 } { 32 }\)
• D) \(\frac { 3 } { 16 }\)

---

Definitions:

Equivalent Units

A measurement used in process costing that converts partial work on units into a number of fully completed units.

First-In, First-Out

An inventory management and valuation method where the oldest inventory items are sold or used first.

Equivalent Units

A concept used in process costing that converts partially completed units into a number of equivalent fully completed units.

Conversion Costs

Costs incurred during the transformation of raw materials into finished goods, typically including labor and overhead expenses.