Infinite series of nth root of n factorial
Consider: $\lim_{n \to \infty} 1 = \lim_{n\to \infty} (1/n + \cdots + 1/n) = \sum \lim_{n \to \infty} 1/n = \sum 0 = 0$, and compare that with what you did. Do you understand why your second "equality" isn't correct?
Your solution: The number of terms is n and n goes to infinity and the multiplication of one infinitely many times is not one.
An easy way to approach it is Stirling's approximation: $n! \approx (\frac ne)^n\sqrt{2 \pi n}$ so $n!^{\frac 1n} \approx \frac ne \to \infty$