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
Steve M
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 #

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