Provide an Appropriate Response $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n } { 6 ^ { n - 1 } }$

question 142

Multiple Choice

Provide an appropriate response.
-Find the sum of the series $\sum _ { n = 1 } ^ { \infty } ( - 1 ) ^ { n - 1 } \frac { n } { 6 ^ { n - 1 } }$ by expressing $\frac { 1 } { 1 + x }$ as a geometric series, differentiating both sides of the resulting equation with respect to $x$ , and replacing $x$ by $\frac { 1 } { 6 }$ .

• A) \(\frac { 49 } { 36 }\)
• B) \(\frac { 36 } { 49 }\)
• C) \(\frac { 36 } { 25 }\)
• D) \(\frac { 25 } { 36 }\)

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Definitions:

Firing

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Nonnegative Integer

An integer value that is greater than or equal to zero, implying it includes all positive integers and zero.

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Zero-Argument Constructor

A zero-argument constructor is a type of constructor method in object-oriented programming that does not take any parameters, used to create objects with default settings or values.