试证明下列命题:
(1) 设 $f \in {C}^{\left( 1\right) }\left( \left[ {0,1}\right] \right)$ ，则 ${B}_{n}^{\prime }\left( {f,x}\right)$ 在 $\left[ {0,1}\right]$ 上一致收敛于 $f'\left( x\right)$ .
(2) 设 $f \in C\left( \left( {0,1}\right) \right)$ (点 $x = 0$ 或1为 $f$ 之瑕点)，且瑕积分 ${\int }_{0}^{1}f\left( x\right) \mathrm{d}x$ 收敛. 若 ${\int }_{0}^{1}f\left( x\right) {x}^{n}\mathop{}\!\mathrm{d}x = \; 0\left( {n = 0,1,2,\cdots }\right)$ ,则 $f\left( x\right) \equiv 0$ .
(3) $I = {\int }_{0}^{+\infty }{x}^{n} \cdot \sin \left( {x}^{1/4}\right) {\mathrm{e}}^{-{x}^{1/4}}\mathop{}\!\mathrm{d}x = 0\quad(n=0,1,2,\cdots)$ .
(4) 设 $f\left( x\right) = \mathop{\sum }\limits_{n = 0}^{\infty }{a}_{n}{x}^{n}$ 的收敛半径为 $1,{a}_{n} > 0\left( {n \in \mathbf{N}}\right)$ 且 $\mathop{\lim }\limits_{{x \rightarrow {1}^{ - }}}\left( {1 - x}\right) f\left( x\right) = 1$ ，则 $\mathop{\lim }\limits_{n \rightarrow \infty }\left( {{a}_{1} + }\right. \; \left. {{a}_{2} + \cdots + {a}_{n}}\right) /n = 1$ .