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Answer:
The usual way of writing the nth root of n is as n1n . In the complex plane, there are n roots for integer n (and infinitely many roots for irrational n !), but let’s focus on the positive real values for positive real arguments.
This is a fun function. Let’s call it f and look at a few values. f(1)=1 . f(2)=2–√ . f(0) … well, that is undefined. As n goes to infinity, f(n) obviously goes to one, and has the asymptotic expansion 1+logn+⋯n+⋯ .
We can find its maximum value by looking at f′(n)=n1n(1n2−lognn2) . This is obviously zero when logn=1 , that is, n=e . The maximum value is e1e=1.44466786⋯ , not an especially interesting number. For integer n, the maximum value is at f(3)=1.44224957⋯ .
Now let’s look at its value near 0. It turns out that it goes to zero very fast for positive arguments: f(10−1)≈10−10 and f(10−10)≈10−100000000000 . But for negative arguments, it blows up: f(−10−1)≈1010 , f(−11−1)≈−2.81011 and f(−10−10)≈10100000000000 . The right limit is 0, but the left limit is undefined, so the value at 0 is undefined.
Things get messy for non-real values… but another possibly interesting value is the real value of f(±i)=e∓π2 .