sequences and series - Find $lim_{nto infty} left(left(1+frac 1{n2^{n+1}}right)^n -1right)*2^n$.

I know the limit is $\frac 12$ and I proved it with sandwich theorem. Does anyone know how to prove it in a constructive way.

$$\lim_{n\to \infty} \left(\left(1+\frac 1{n2^{n+1}}\right)^n -1\right)*2^n$$

Answer

Hint $:$ Expand binomially

$$\left (1 + \frac {1} {n2^n} \right )^n.$$

You will see that all the terms of $\left(\left(1+\frac 1{n2^{n+1}}\right)^n -1\right)*2^n$ will vanish when $n \rightarrow \infty$ except that $\frac 1 2.$