How do you determine if $a_{n} = {(1 + n)}^{\frac{1}{n}}$ $a_n=(1+n)^(1/n)$ converge and find the limits when they exist?

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Answer 1 · 2017-11-08T15:00:30

The series converge to $= 1$ $=1$

Let $y = {(1 + n)}^{\frac{1}{n}}$ $y=(1+n)^(1/n)$

Taking logarithms

$ln y = ln ({(1 + n)}^{\frac{1}{n}}) = \frac{1}{n} ln (1 + n)$ $lny=ln((1+n)^(1/n))=1/nln(1+n)$

Therefore,

$y = e^{\frac{1}{n} ln (1 + n)}$ $y=e^(1/nln(1+n))$

So,

$lim_(n->oo)u_n=lim_(n->oo)e^(1/nln(1+n))$

Apply the limit chain rule

$lim_(n->oo)(1/nln(1+n))=lim_(n->oo)(1/(1+n))$ ( by L'Hôpital's rule)

$=0$

Therefore,

$lim_(n->oo)e^(1/nln(1+n))=lim_(n->oo)e^0=1$

How do you determine if a_n=(1+n)^(1/n)an=(1+n)1na_n=(1+n)^(1/n) converge and find the limits when they exist?