91Ó°ÊÓ

In each of Problems 1 through 10 test for convergence or divergence. $$ \sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{n^{2}}} $$

In each of Problems 1 through 10 show that the sequence \(\left\\{f_{n}(x)\right\\}\) converges to \(f(x)\) for each \(x\) on \(I\) and determine whether or not the convergence is uniform. $$ f_{n}: x \rightarrow \frac{n^{3} x}{1+n^{4} x^{2}}, \quad f(x) \equiv 0, \quad I=\\{x: a \leqslant x<\infty, a>0\\} $$

Show that the sequence \(f_{n}: x \rightarrow x^{n}\) converges for each \(x \in I=\\{x: 0 \leqslant x \leqslant 1\\}\) but that the convergence is not uniform.