I'm puzzled with the following limit: and

the exponent is

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

Using online graph plotter I can see that the answer is but no matter what I try I can't reach this answer analytically.

I tried quite a few manipulations like: extrating factor, geometric sum, representing the formula recursively through and and some other symbol manipulations.

Can someone point me to the promising direction ?

Thank you