Does the series a_n=(1+n)^(1/n) converge or diverge?

use L'Hospital's Rule
(which I'm terrible at :( )

I tried to find the limit but got stuck...

1 Answer
Apr 5, 2018

lim_(n rarr oo) (1+n)^(1/n) = 1

Explanation:

If You are applying L'Hôpital's rule , then it is assumed that we seek:

L = lim_(n rarr oo) (1+n)^(1/n)

We can take Natural logarithms:

ln L = ln {lim_(n rarr oo) (1+n)^(1/n)}

Using the monotonicity of the logarithmic function we can write:

ln L = lim_(n rarr oo) {ln(1+n)^(1/n) }

Then using the properties of logarithms:

ln L = lim_(n rarr oo) {1/n ln(1+n) }

\ \ \ \ \ \ \ = lim_(n rarr oo) { (ln(1+n))/n }

This limit is of an indeterminate form oo//oo, so we can apply L'Hôpital's rule to get:

ln L = lim_(n rarr oo) { (d/(dn) ln(1+n))/(d/(dn) n) }

\ \ \ \ \ \ \ = lim_(n rarr oo) (1/(1+n))/1

\ \ \ \ \ \ \ = lim_(n rarr oo) 1/(1+n)

\ \ \ \ \ \ \ = 0

And so:

L = e^0 = 1