1. ## Convergence

N is a positive number.

How can i show that $N^{1/n}(1+n)^{1/n}$ tends to $1$ as $n$ tends to $\infty.$

2. Hi mate,

notice how $1/n \rightarrow 0$ as $n \rightarrow \infty$.
You know, $1/2=0.5, 1/3=0.33, 1/10=0.1, 1/100=0.01$ etc.

Then $N^{1/n}$ tends to $1$ and so does $(1+n)^{1/n}$, because we know that a number to the zeroth power is 1.

Hope that makes sense.

3. Clearly $N^{1/n}$ tends to $1.$ But the problem is with $(1+n)^{1/n}$ because you get $\infty^{0}$which doesnot make sense.

4. Is N a constant then, and not just "n"?

If $y= (1+ n)^{1/n}$ 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 $(1+ n)^{1/n}$ is, indeed, 1.