Find a Formula for the Nth Partial Sum of the Series

question 36

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.
- $$3 - \frac { 3 } { 7 } + \frac { 3 } { 49 } - \frac { 3 } { 343 } + \ldots + ( - 1 ) ^ { n - 1 } \frac { 3 } { 7 ^ { n - 1 } } + \ldots$$

• A) \(\frac { 3 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 7 } { 2 }\)
• B) \(\frac { 3 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n - 1 } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 7 } { 2 }\)
• C) \(\frac { 3 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 21 } { 8 }\)
• D) \(\frac { 3 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n - 1 } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 21 } { 8 }\)

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Definitions:

Formal Hierarchy

An organizational structure where each position and its authority relative to others is clearly outlined, establishing a clear chain of command.

Authority

The legitimate power granted to individuals or groups, typically within an organizational structure, to make decisions and enforce rules.

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The influence gained from networking or building relationships with influential individuals or groups.

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The interconnected system of nodes (which can be individuals, organizations, or computers) that allows for the exchange of information, resources, or services.