Find a Formula for the Nth Partial Sum of the Series

question 346

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.
- $9 - 36 + 144 - 576 + \ldots + ( - 1 ) ^ { n - 1 } 9 \cdot 4 ^ { n - 1 } + \ldots$

• A) \(\frac { 9 \left( 1 - ( - 4 ) ^ { n } \right) } { 4 - 1 }\); series diverges
• B) \(\frac { 9 \left( 1 - ( - 4 ) ^ { n } \right) } { 1 + 4 }\); series diverges
• C) \(\frac { 9 \left( 1 - ( - 4 ) ^ { n } \right) } { 1 + 4 } ; \frac { 9 } { 5 }\)
• D) \(\frac { 9 \left( 1 - ( - 4 ) ^ { n } \right) } { 1 - 4 } ; 3\)

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Definitions:

Speech Sound Disorder

A condition where a person has difficulty producing or pronouncing sounds correctly, affecting verbal communication.

Articulation

The physical production of particular speech sounds and the clarity with which these sounds are produced in spoken language.

Childhood Onset Fluency

A speech disorder originating in childhood that affects the flow, speed, and rhythm of speech, often referred to as stuttering.

Stutters

A speech disorder characterized by repeated or prolonged sounds, syllables, or words, disrupting the normal flow of speech.