Find a Formula for the Nth Partial Sum of the Series

question 63

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.
- $$5 - \frac { 5 } { 7 } + \frac { 5 } { 49 } - \frac { 5 } { 343 } + \ldots + ( - 1 ) ^ { n - 1 } \frac { 5 } { 7 ^ { n - 1 } } + \ldots$$

• A) \(\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n - 1 } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 6 }\)
• B) \(\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 6 }\)
• C) \(\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n - 1 } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 8 }\)
• D) \(\frac { 5 \left( 1 - \frac { 1 } { ( - 7 ) ^ { n } } \right) } { 1 + \frac { 1 } { 7 } } ; \frac { 35 } { 8 }\)

---

Definitions:

Hallucinogens

Drugs that powerfully change perception and can trigger sensory experiences in the absence of any inputs.

Stimulants

Substances that increase physiological or nervous activity in the body, often used to enhance alertness, attention, and energy.

Depressants

Drugs that diminish activity levels in the nervous system.

Nose Drops

A liquid medication intended for administration in the nose to alleviate symptoms such as nasal congestion.