Determine the limit of the sequence $a _ { n } = ( - 1 ) ^ { n } \left( 1 - \frac { 1 } { \sqrt { n } } \right)$ .
A)-2
B)0
C)ln 2
D) $\sqrt { 2 }$
E) $e ^ { 2 }$
F) $- \frac { 1 } { 2 }$
G)1

H)Divergent

Question

Determine the limit of the sequence $a _ { n } = ( - 1 ) ^ { n } \left( 1 - \frac { 1 } { \sqrt { n } } \right)$ .
A)-2
B)0
C)ln 2
D) $\sqrt { 2 }$ 
E) $e ^ { 2 }$ 
F) $- \frac { 1 } { 2 }$ 
G)1

H)Divergent

Quizplus · Accepted Answer

The sequence $a_n = (-1)^n\left(1 - \frac{1}{\sqrt{n}}\right)$ oscillates between positive and negative values as $n$ increases, and it does not approach a single value. The factor $(-1)^n$ causes the sequence to alternate signs, and although $\left(1 - \frac{1}{\sqrt{n}}\right)$ approaches 1 as $n$ goes to infinity, the alternating sign prevents the sequence from converging to a specific limit. Therefore, the sequence is divergent.

Determine the Limit of the Sequence an=(−1)n(1−1n)a _ { n } = ( - 1 ) ^ { n } \left( 1 - \frac { 1 } { \sqrt { n } } \right)an​=(−1)n(1−n​1​)

Determine the Limit of the Sequence $a _ { n } = ( - 1 ) ^ { n } \left( 1 - \frac { 1 } { \sqrt { n } } \right)$