Proving $\lim n^{\frac{1}{n}} =1.$

Limit of $\sum \dfrac{1}{1^2+2^2+\cdots+n^2}$ Number of trials rolling six 6-sided dice to get 6 unique values Calculus Critical Point Confusion - 2nd Derivative Confusion Solve $2 \cos^2 x+ \sin x=1$ for all possible $x$ Solve a nonlinear differential equation of the first order Formalizing a proof for $ \sum_{n=0}^\infty \sum_{k=n}^{\infty} a_k = \sum_{n=0}^{\infty} (n+1)a_n$ Quadratic subfield of $\mathbb{Q}(\zeta)$ Prove that $\sum_{i=1}^{n}\frac{1}{\left(n+i\right)^{2}}\sim\frac1{2n}$ Riemann sum of $\int_1^2 {1\over x^2} dx$. Is there a short, elementary way to show $\cos(x+y)=\cos x\cos y-\sin x\sin y$ for any angles $x$ and $y$? Evaluating $\lim_{x\to0}{\frac{\sqrt[3]{\cos{4x}}-\sqrt[3]{\cos{5x}}}{1-\cos{3x}}}$ without L'Hopital's rule is log 0 is or is not undefined?