Question

Determine the limit of the following sequence, or state that it diverges.

Answer 1

Hey Alb :)
As n approaches infinity,\[\large\rm \lim_{n\to\infty}n^{3/n}=(\infty)^{0}\]It looks like we're getting an indeterminate form, ya?
So we'll have to try something clever...

Answer 2

Recall that the exponential and log functions are inverses of one another.
When we take their composition, we get back the argument,\[\large\rm \color{orangered}{x}=e^{\ln(\color{orangered}{x})}\]

Answer 3

We're going to apply this to our limit problem here,\[\large\rm \color{orangered}{\lim_{n\to\infty}n^{3/n}}=e^{\ln(\color{orangered}{\lim_{n\to\infty}n^{3/n}})}\]Ignore the exponential base for a moment,
focus on what's happening in the exponent.
Pass the limit outside of the log,\[\large\rm \ln\left(\lim_{n\to\infty}n^{3/n}\right)=\lim_{n\to\infty} \ln(n^{3/n})\]

Answer 4

Apply log rule: \(\large\rm \log(a^b)=b\cdot\log(a)\)

Answer 5

\[\large\rm \lim_{n\to\infty}\frac{3}{n}\cdot\ln(n)\quad=\quad 3\lim_{n\to\infty}\frac{\ln(n)}{n}\]

Answer 6

Notice that we're STILL getting an indeterminate form, \(\large\rm \frac{\infty}{\infty}\)
But it's one of those special indeterminate forms!
We can apply L'Hospital's Rule! Yay!

Answer 7

Albeartooooooo, following any of this? :)
Whatchu think?

Answer 8

Yeah, im following down everything. L'Hospital is done on ln(n)/n ?

Answer 9

Yes. What do we get for our derivatives? :)

Answer 10

ln(n)/1---> (1/n)/1

Answer 11

Ok cool, let's simplify it:\[\large\rm 3\lim_{n\to\infty}\frac{\frac{1}{n}}{1}\quad=\quad3\lim_{n\to\infty}\frac{1}{n}\]So how bout now?
Are we still getting indeterminate form or no? :o

Answer 12

no lol as n goes to infinity, we get close to zero. so, 3 *0= 0?

Answer 13

Ok great.
And now let's not forget about our exponential base!!!
Remember, we were doing all of this within our exponent.

Answer 14

the limit is zero!

Answer 15

ooooohhh

Answer 16

so, e^(3*0). it equals one!

Answer 17

\[\large\rm e^{\color{royalblue}{\lim_{n\to\infty} \ln(n^{3/n})}}\quad=\quad e^{\color{royalblue}{0}}\]

So we determined that it converges!
Awesome! :)

Answer 18

Yay team! ୧ʕ•̀ᴥ•́ʔ୨

Answer 19

Haha! you're awesome! thank you for your help. i need to relearn these formulas again, or study them.

Answer 20

ya lots of weird rules to remember D: