Find the limit of the sequence $a _ { n } = \left( n + e ^ { n } \right) ^ { 1 / n }$ .
A)1
B)2
C) $e$
D) $1 + e$
E)-1
F)3
G)Divergent

Question

Find the limit of the sequence $a _ { n } = \left( n + e ^ { n } \right) ^ { 1 / n }$ .
A)1
B)2
C) $e$ 
D) $1 + e$ 
E)-1
F)3
G)Divergent

Quizplus · Accepted Answer

The limit of the sequence $a_n = (n + e^n)^{1/n}$ can be found by recognizing that as $n$ approaches infinity, the term $e^n$ grows much faster than $n$, making it the dominant term in the expression. Therefore, the sequence behaves similarly to $(e^n)^{1/n}$, which simplifies to $e$. Thus, the limit is $e$.

Find the Limit of the Sequence an=(n+en)1/na _ { n } = \left( n + e ^ { n } \right) ^ { 1 / n }an​=(n+en)1/n

Find the Limit of the Sequence $a _ { n } = \left( n + e ^ { n } \right) ^ { 1 / n }$