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**ktotam** I'm puzzled with the following limit: $\displaystyle \lim_{n \rightarrow \infty } {\sqrt[n]{{b}^{ \frac{1}{ 2^{n} } } - 1}} $ and $\displaystyle b > 1$

the exponent is $\displaystyle \frac{1}{2^n}$

It is easy to see that the limit lies in [0,1] interval.

Using online graph plotter I can see that the answer is $\displaystyle \frac{1}{2}$ but no matter what I try I can't reach this answer analytically.

I tried quite a few manipulations like: extrating $\displaystyle \frac{1}{2^{n}}$ factor, geometric sum, representing the formula recursively through $\displaystyle a_n$ and $\displaystyle a_{n+1}$ and some other symbol manipulations.

Can someone point me to the promising direction ?

Thank you