1. n^(1/n)

Prove that the series $\displaystyle a_n=\root n\of {n}$ starting with $\displaystyle n=1$ has its maximum at $\displaystyle n=3$, therefore afterwords it strictly decreasing. (I know that $\displaystyle \lim_{n\rightarrow\infty}\root n\of {n}=1$.)

2. Hi

Maybe there is another way but I would do it by studying the function $\displaystyle f(x) = x^{\frac{1}{x}} = e^{\frac{ln x}{x}}$

3. Hi I'm new here. Just curious what level of math this would be??

4. I cannot really answer your question since I don't know the American education system.

What I can say is that you need to know how to calculate the derivative of $\displaystyle e^{\frac{ln x}{x}}$.

5. $\displaystyle f'(x)=e^{\frac {\ln x}x} \left(\frac{1}{x^2}-\frac{\ln x}{x^2}\right)=0\rightarrow x=e$. Thanks!

6. Fine !

Now you need to compare $\displaystyle 2^{\frac{1}{2}}$ and $\displaystyle 3^{\frac{1}{3}}$

7. Originally Posted by becky129622
Hi I'm new here. Just curious what level of math this would be??
Probably 1st year university level (but it probably depends on the country of origin).