📝 题目
例 7 设 ${u}_{n}\left( x\right) \in C\left\lbrack {a,b}\right\rbrack \left( {n = 1,2,\cdots }\right)$ ,级数 $\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\left( x\right)$ 在(a, b) 上一致收敛. 求证:
(1) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\left( a\right) ,\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\left( b\right)$ 收敛;
(2) $\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\left( x\right)$ 在 $\left\lbrack {a,b}\right\rbrack$ 上一致收敛.
💡 答案与解析
证(1) $\forall \varepsilon > 0$ ,由条件 $\exists N \in N$ ,当 $n > N$ ,对 $\forall p \in N$ ,
$$ \left| {{u}_{n + 1}\left( x\right) + {u}_{n + 2}\left( x\right) + \cdots + {u}_{n + p}\left( x\right) }\right| < \varepsilon \;\left( {\forall x \in \left( {a,b}\right) }\right) , $$
令 $x \rightarrow a + 0$ ,得
$$ \left| {{u}_{n + 1}\left( a\right) + {u}_{n + 2}\left( a\right) + \cdots + {u}_{n + p}\left( a\right) }\right| \leq \varepsilon . $$
由数值级数收敛原理知 $\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\left( a\right)$ 收敛. 同理可证 $\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\left( b\right)$ 收敛.
(2)由(1)的证明过程可知,对 $\forall \varepsilon > 0$ , $\exists N \in N$ ,当 $n > N$ 时,对 $\forall p \in N$ ,有
$$ \left| {{u}_{n + 1}\left( x\right) + {u}_{n + 2}\left( x\right) + \cdots + {u}_{n + p}\left( x\right) }\right| \leq \varepsilon \;\left( {\forall x \in \left\lbrack {a,b}\right\rbrack }\right) . $$
由此可见, $\mathop{\sum }\limits_{{n = 1}}^{\infty }{u}_{n}\left( x\right)$ 在 $\left\lbrack {a,b}\right\rbrack$ 上一致收敛.