Answer
The sequence converges,
$\displaystyle \lim_{n\rightarrow\infty}a_{n}=1$
Work Step by Step
Th.9.1:
"Let $L$ be a real number.
Let $f$ be a function of a real variable such that $\displaystyle \lim_{x\rightarrow\infty}f(x)=L$.
If $\{a_{n}\}$ is a sequence such that $f(n)=a_{n}$ for every positive integer $n$,
then $\displaystyle \lim_{n\rightarrow\infty}a_{n}=L$. "
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We define $f(x)=2^{1/x}$, for which$ f(n)=a_{n}$ and
because $\displaystyle \lim_{x\rightarrow\infty}\frac{1}{x}=0,$
$\displaystyle \lim_{x\rightarrow\infty}f(x)=2^{0}=1.$
Thus,
$\displaystyle \lim_{n\rightarrow\infty}a_{n}=\lim_{x\rightarrow\infty}f(x)=1$.
