How to prove that the sequence $n^{1/n}$ converges to 1

Question

Help me in proving that the sequence $n^{1/n}$ converges to $1$

Answer 1

Hint:

The first thing you want to prove/think about: if $\ln(a_n)\rightarrow L$ as $n\rightarrow\infty$, then $a_n\rightarrow e^L$. You can do this using the continuity of the function $x\mapsto e^x$.

Now, remember that $$ \ln(n^{1/n})=\frac{1}{n}\ln(n). $$ Can you prove that this converges to 0? If so, then the combination of the two facts tells us that $n^{1/n}\rightarrow e^0=1$.