N is a positive number.
How can i show that $\displaystyle N^{1/n}(1+n)^{1/n}$ tends to $\displaystyle 1$ as $\displaystyle n$ tends to $\displaystyle \infty.$
Hi mate,
notice how $\displaystyle 1/n \rightarrow 0$ as $\displaystyle n \rightarrow \infty$.
You know, $\displaystyle 1/2=0.5, 1/3=0.33, 1/10=0.1, 1/100=0.01$ etc.
Then $\displaystyle N^{1/n}$ tends to $\displaystyle 1$ and so does $\displaystyle (1+n)^{1/n}$, because we know that a number to the zeroth power is 1.
Hope that makes sense.
Is N a constant then, and not just "n"?
If $\displaystyle y= (1+ n)^{1/n}$ then ln(y)= ln(1+ n)/n. By L'Hopital's rule, ln(1+x)/x goes to 0 as x goes to infinity so ln(1+ n)/n goes to 0 as n goes to infinity. Since y is 0, the limit of $\displaystyle (1+ n)^{1/n}$ is, indeed, 1.